{ "generated": "2026-11-10 22:56 UTC", "curves": [ { "schema": "genus-2-torsion/curve/1", "id": "1.a", "group": [], "group_key": "1", "group_bracket": "[]", "order": 1, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "-14580", "51520", "-63684", "31054", "-3823", "-637", "-18" ], "h": [ "1" ], "f_str": "-18*x^6 - 637*x^5 - 3823*x^4 + 31054*x^3 - 63684*x^2 + 51520*x - 14580", "h_str": "1", "g": [ "-58319", "206080", "-254736", "124216", "-15292", "-2548", "-72" ], "g_str": "-72*x^6 - 2548*x^5 - 15292*x^4 + 124216*x^3 - 254736*x^2 + 206080*x - 58319", "g_scale": "1", "discriminant": "4917915233292686239789080727552", "discriminant_digits": 31, "igusa_clebsch": [ "8239495424", "3773859343832050432", "7920670268304843944401723392", "4917915233292686239789080727552" ], "g2_invariants": [ "1158919177308540097746397628967835598080442368/1200662898753097226511006037", "5339960680720614867515477384697026489090048/1200662898753097226511006037", "25954357258004485839959272636679603879936/1200662898753097226511006037" ], "rational_weierstrass_points": 1, "minimal_model": { "f": [ "-18", "745", "-7278", "-9032", "-100", "1271", "-168" ], "h": [ "0", "0", "0", "1" ] }, "minimal_discriminant": "1200662898753097226511006037", "minimal_discriminant_factored": "113^12 * 277", "conductor": "277", "conductor_factored": "277", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "277.a.3", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-14580%2C51520%2C-63684%2C31054%2C-3823%2C-637%2C-18%5D%2C%5B1%5D%5D", "snapshot_label": "277.a.3" }, "torsion": { "invariants": [], "order": 1, "generators": [], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 15, "point_count_primes": [ 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 7, "chi": "T^4 - T^3 + 3*T^2 - 7*T + 49", "primes_tried_below": 500, "admissible_primes": 2, "reducible_chi": 1 }, "q_simple": { "certified": true, "prime": 7, "chi": "T^4 - T^3 + 3*T^2 - 7*T + 49" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 43, "reducible_chi": 13 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 1 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-14580", "51520", "-63684", "31054", "-3823", "-637", "-18" ], "h": [ "1" ], "claimed_group": null, "claimed_class": "simple", "cover": null, "lmfdb_label_given": "277.a.3" }, "source": { "kind": "import", "name": "paper-t1-001", "file": "table1.txt", "line": 7, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:10 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.330000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/1.a.log", "conductor_log": "data/logs/1.a.conductor.log", "conductor_cputime_seconds": 0.16 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "1.b", "group": [], "group_key": "1", "group_bracket": "[]", "order": 1, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-6", "-33", "-14", "95", "-50", "3", "-1" ], "h": [ "1" ], "f_str": "-x^6 + 3*x^5 - 50*x^4 + 95*x^3 - 14*x^2 - 33*x - 6", "h_str": "1", "g": [ "-23", "-132", "-56", "380", "-200", "12", "-4" ], "g_str": "-4*x^6 + 12*x^5 - 200*x^4 + 380*x^3 - 56*x^2 - 132*x - 23", "g_scale": "1", "discriminant": "-1601384448", "discriminant_digits": 10, "igusa_clebsch": [ "601760", "31128254272", "4413239365215232", "-1601384448" ], "g2_invariants": [ "-2408056349828975363200000/390963", "1982406707133537344000/20577", "-27053302090985600/19" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "-6", "33", "-14", "-95", "-50", "-3", "-1" ], "h": [ "1" ] }, "minimal_discriminant": "-390963", "minimal_discriminant_factored": "-1 * 3 * 19^4", "conductor": "1083", "conductor_factored": "57 * 19 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 57 * 19 for the elliptic curves ['0', '1', '1', '-4390', '-113432'] and ['0', '1', '1', '-769', '-8470'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "1083" }, "lmfdb": { "label": "1083.b.390963.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/1083/b/390963/1", "snapshot_label": "" }, "torsion": { "invariants": [], "order": 1, "generators": [], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 15, "point_count_primes": [ 5, 7, 11, 13, 17, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x - z : y + z^3 : -z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "0", "1", "1", "-4390", "-113432" ], [ "0", "1", "1", "-769", "-8470" ] ], "conductors": [ "57", "19" ], "conductor": "1083", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 1 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-6", "-33", "-14", "95", "-50", "3", "-1" ], "h": [ "1" ], "claimed_group": null, "claimed_class": "split", "cover": null, "lmfdb_label_given": "1083.b.390963.1" }, "source": { "kind": "import", "name": "paper-t2-001", "file": "table2.txt", "line": 7, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:33:14 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.11000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/1.b.log", "conductor_log": "data/logs/1.b.conductor.log", "conductor_cputime_seconds": 0.22 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "1.c", "group": [], "group_key": "1", "group_bracket": "[]", "order": 1, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-7", "-37", "-122", "-129", "-42", "-5", "-7" ], "h": [ "0" ], "f_str": "-7*x^6 - 5*x^5 - 42*x^4 - 129*x^3 - 122*x^2 - 37*x - 7", "h_str": "0", "g": [ "-28", "-148", "-488", "-516", "-168", "-20", "-28" ], "g_str": "-28*x^6 - 20*x^5 - 168*x^4 - 516*x^3 - 488*x^2 - 148*x - 28", "g_scale": "1", "discriminant": "-121482269760679367933952", "discriminant_digits": 24, "igusa_clebsch": [ "216032", "33700446208", "2007194214072320", "-121482269760679367933952" ], "g2_invariants": [ "-14022987275143158751/28963630142373888", "7518863711686187/35316116619264", "639932630041/32296402944" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "-7", "-37", "-122", "-129", "-42", "-5", "-7" ], "h": [] }, "minimal_discriminant": "-29658757265790861312", "minimal_discriminant_factored": "-1 * 2^24 * 3^21 * 13^2", "conductor": "1521", "conductor_factored": "3^2 * 13^2", "conductor_note": "taken from the LMFDB label 1521.d.2 (Magma's genus-2 Conductor failed on this curve: Fibre blowups over number fields not yet implemented)", "conductor_source": "lmfdb label" }, "lmfdb": { "label": "1521.d.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-7%2C-37%2C-122%2C-129%2C-42%2C-5%2C-7%5D%2C%5B0%5D%5D", "snapshot_label": "1521.d.2" }, "torsion": { "invariants": [], "order": 1, "generators": [], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 7, "point_count_primes": [ 5, 11, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 91, "reducible_chi": 25 }, "q_simple": { "certified": true, "prime": 5, "chi": "T^4 + 2*T^2 + 25" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "geometric_involution", "over": "Qbar", "detail": "the geometric automorphism group has order 12 (7 involutions), so the curve is bielliptic over Qbar" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 42, "reducible_chi": 13 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": 2026, "historical": false }, "reference": "LMFDB extended genus 2 database (Booker–Sutherland), curve 1521.d.2 (table g2c_curves_new, label 1521.d2 on devmirror.lmfdb.xyz, read 2026-10-16)", "sources": [ "BookerSutherland" ], "rm": false, "notes": "First Q-simple, geometrically split example of [] for the census (the gsplit cell is empty). Found by querying the LMFDB genus 2 data (devmirror.lmfdb.xyz, g2c_curves_new: is_simple_base = true, is_simple_geom = false, torsion_subgroup = []; geom_end_alg = M_2(Q), end_alg = CM; smallest conductor among the curves of this group that the census verifier certifies). LMFDB conductor 1521. Checked locally with the census's own pipeline/magma/verify_lib.m (Magma V2.30-1): J(Q)_tors = [], class gsplit, Q-simplicity prime 5 (chi = T^4 + 2*T^2 + 25), splitness certificate: geometric_involution over Qbar (the geometric automorphism group has order 12 (7 involutions), so the curve is bielliptic over Qbar). Submitted by F. Najman; the curve is due to the database cited under Reference.", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "-7", "-37", "-122", "-129", "-42", "-5", "-7" ], "h": [ "0" ], "claimed_group": null, "claimed_class": "split", "cover": null, "lmfdb_label_given": "1521.d.2" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_1.json" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-17 06:01:47 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "0.860000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/1.c.log", "conductor_log": "data/logs/1.c.conductor.log", "conductor_cputime_seconds": null }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.a", "group": [ 2 ], "group_key": "2", "group_bracket": "[2]", "order": 2, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "-608", "389", "22", "-40", "0", "1" ], "h": [ "1", "1", "1" ], "f_str": "x^5 - 40*x^3 + 22*x^2 + 389*x - 608", "h_str": "x^2 + x + 1", "g": [ "-2431", "1558", "91", "-158", "1", "4" ], "g_str": "4*x^5 + x^4 - 158*x^3 + 91*x^2 + 1558*x - 2431", "g_scale": "1", "discriminant": "-75520", "discriminant_digits": 5, "igusa_clebsch": [ "397608", "1223228004", "147861032449512", "-1208320" ], "g2_invariants": [ "-303267334973269931148501/295", "-2214359494206283568520/59", "-502441543825401014496/295" ], "rational_weierstrass_points": 1, "minimal_model": { "f": [ "-18", "94", "-129", "2", "10", "1" ], "h": [ "1", "1", "1" ] }, "minimal_discriminant": "-295", "minimal_discriminant_factored": "-1 * 5 * 59", "conductor": "295", "conductor_factored": "5 * 59", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "295.a.295.2", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/295/a/295/2", "snapshot_label": "" }, "torsion": { "invariants": [ 2 ], "order": 2, "generators": [ { "order": 2, "mumford": [ "x^2 - 7*x + 11", "0", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 14, "point_count_primes": [ 3, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 2 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 7, "chi": "T^4 - T^3 - 7*T + 49", "primes_tried_below": 500, "admissible_primes": 2, "reducible_chi": 1 }, "q_simple": { "certified": true, "prime": 7, "chi": "T^4 - T^3 - 7*T + 49" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 43, "reducible_chi": 8 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 2 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-608", "389", "22", "-40", "0", "1" ], "h": [ "1", "1", "1" ], "claimed_group": [ 2 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "295.a.295.2" }, "source": { "kind": "import", "name": "paper-t1-002", "file": "table1.txt", "line": 8, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:11 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.260000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.a.log", "conductor_log": "data/logs/2.a.conductor.log", "conductor_cputime_seconds": 0.11 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.b", "group": [ 2 ], "group_key": "2", "group_bracket": "[2]", "order": 2, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-56", "0", "-75", "0", "15", "0", "-1" ], "h": [ "0", "1", "0", "1" ], "f_str": "-x^6 + 15*x^4 - 75*x^2 - 56", "h_str": "x^3 + x", "g": [ "-224", "0", "-299", "0", "62", "0", "-3" ], "g_str": "-3*x^6 + 62*x^4 - 299*x^2 - 224", "g_scale": "1", "discriminant": "-704643072", "discriminant_digits": 9, "igusa_clebsch": [ "135328", "9671540800", "119230681068032", "-704643072" ], "g2_invariants": [ "-1352659309173012149/168", "419870026410625699/168", "-461744933079368" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "-56", "0", "-75", "0", "15", "0", "-1" ], "h": [ "0", "1", "0", "1" ] }, "minimal_discriminant": "-172032", "minimal_discriminant_factored": "-1 * 2^13 * 3 * 7", "conductor": "336", "conductor_factored": "14 * 24 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 14 * 24 for the elliptic curves ['1', '0', '1', '-2731', '-55146'] and ['0', '-1', '0', '-384', '-2772'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "336" }, "lmfdb": { "label": "336.a.172032.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/336/a/172032/1", "snapshot_label": "" }, "torsion": { "invariants": [ 2 ], "order": 2, "generators": [ { "order": 2, "mumford": [ "x^2 - 32/3", "0", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 48, "point_count_primes": [ 5, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 2 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x : y : -z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "0", "1", "-2731", "-55146" ], [ "0", "-1", "0", "-384", "-2772" ] ], "conductors": [ "14", "24" ], "conductor": "336", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 2 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-56", "0", "-75", "0", "15", "0", "-1" ], "h": [ "0", "1", "0", "1" ], "claimed_group": [ 2 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "336.a.172032.1" }, "source": { "kind": "import", "name": "paper-t2-002", "file": "table2.txt", "line": 8, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:33:15 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.13000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.b.log", "conductor_log": "data/logs/2.b.conductor.log", "conductor_cputime_seconds": 0.41 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.c", "group": [ 2 ], "group_key": "2", "group_bracket": "[2]", "order": 2, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-168", "-666", "-504", "-1", "-504", "665", "-168" ], "h": [ "1", "1", "1", "1" ], "f_str": "-168*x^6 + 665*x^5 - 504*x^4 - x^3 - 504*x^2 - 666*x - 168", "h_str": "x^3 + x^2 + x + 1", "g": [ "-671", "-2662", "-2013", "0", "-2013", "2662", "-671" ], "g_str": "-671*x^6 + 2662*x^5 - 2013*x^4 - 2013*x^2 - 2662*x - 671", "g_scale": "1", "discriminant": "-1260060920654987540294662023806976", "discriminant_digits": 34, "igusa_clebsch": [ "-456342304", "3333764193185536", "-564439887031920685412352", "-1260060920654987540294662023806976" ], "g2_invariants": [ "363828648557660045879696912/185321057216998329", "11276601591690962334757870/185321057216998329", "-810094579498129/3138428376721" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "-168", "-666", "-504", "-1", "-504", "665", "-168" ], "h": [ "1", "1", "1", "1" ] }, "minimal_discriminant": "-307632060706784067454751470656", "minimal_discriminant_factored": "-1 * 2^6 * 3^10 * 11^22", "conductor": "576", "conductor_factored": "2^6 * 3^2", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "576.b.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-168%2C-666%2C-504%2C-1%2C-504%2C665%2C-168%5D%2C%5B1%2C1%2C1%2C1%5D%5D", "snapshot_label": "576.b.2" }, "torsion": { "invariants": [ 2 ], "order": 2, "generators": [ { "order": 2, "mumford": [ "x^2 + 1", "0", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "the submitted generators generate a subgroup of order 2, but the point-count bound is 20; TorsionSubgroup was run", "generators_submitted": 1, "point_count_gcd": 20, "point_count_primes": [ 5, 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 2 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 91, "reducible_chi": 54 }, "q_simple": { "certified": true, "prime": 19, "chi": "T^4 - 22*T^2 + 361" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "richelot", "over": "Q", "detail": "(2,2)-isogenous over Q to a Weil restriction of an elliptic curve E over Q(sqrt(2)) (the surface is E x E^sigma)" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 42, "reducible_chi": 24 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": 2026, "historical": false }, "reference": "LMFDB extended genus 2 database (Booker–Sutherland), curve 576.b.2 (table g2c_curves_new, label 576.b2 on devmirror.lmfdb.xyz, read 2026-10-16)", "sources": [ "BookerSutherland" ], "rm": false, "notes": "First Q-simple, geometrically split example of [2] for the census (the gsplit cell is empty). Found by querying the LMFDB genus 2 data (devmirror.lmfdb.xyz, g2c_curves_new: is_simple_base = true, is_simple_geom = false, torsion_subgroup = [2]; geom_end_alg = M_2(Q), end_alg = CM; smallest conductor among the curves of this group that the census verifier certifies). LMFDB conductor 576. Checked locally with the census's own pipeline/magma/verify_lib.m (Magma V2.30-1): J(Q)_tors = [2], class gsplit, Q-simplicity prime 19 (chi = T^4 - 22*T^2 + 361), splitness certificate: richelot over Q ((2,2)-isogenous over Q to a Weil restriction of an elliptic curve E over Q(sqrt(2)) (the surface is E x E^sigma)). Submitted by F. Najman; the curve is due to the database cited under Reference. Rigorous conductor: J is (2,2)-isogenous over Q to Res_(K/Q) E with K the quadratic field of discriminant 8, so N(J) = |d_K|^2 * Norm(f_E) = 64 * 9 = 576 (computed with Magma; conductor is an isogeny invariant).", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "-168", "-666", "-504", "-1", "-504", "665", "-168" ], "h": [ "1", "1", "1", "1" ], "claimed_group": [ 2 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "576.b.2" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_2.json" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-17 06:12:15 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "1.51000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.c.log", "conductor_log": "data/logs/2.c.conductor.log", "conductor_cputime_seconds": 0.37 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "3.a", "group": [ 3 ], "group_key": "3", "group_bracket": "[3]", "order": 3, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "0", "-1", "2", "-2", "1" ], "h": [ "1" ], "f_str": "x^5 - 2*x^4 + 2*x^3 - x^2", "h_str": "1", "g": [ "1", "0", "-4", "8", "-8", "4" ], "g_str": "4*x^5 - 8*x^4 + 8*x^3 - 4*x^2 + 1", "g_scale": "1", "discriminant": "255232", "discriminant_digits": 6, "igusa_clebsch": [ "-128", "256", "107520", "4083712" ], "g2_invariants": [ "-1048576/997", "-32768/997", "-53248/997" ], "rational_weierstrass_points": 1, "minimal_model": { "f": [ "0", "-1", "-2", "-2", "-1" ], "h": [ "0", "0", "0", "1" ] }, "minimal_discriminant": "997", "minimal_discriminant_factored": "997", "conductor": "997", "conductor_factored": "997", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "997.b.997.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/997/b/997/1", "snapshot_label": "" }, "torsion": { "invariants": [ 3 ], "order": 3, "generators": [ { "order": 3, "mumford": [ "x^2 - x", "1", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 3, "point_count_primes": [ 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101 ], "claimed": [ 3 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 3, "chi": "T^4 + 3*T^3 + 5*T^2 + 9*T + 9", "primes_tried_below": 500, "admissible_primes": 1, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 + 3*T^3 + 5*T^2 + 9*T + 9" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 45, "reducible_chi": 11 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 3 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "0", "-1", "2", "-2", "1" ], "h": [ "1" ], "claimed_group": [ 3 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "997.b.997.1" }, "source": { "kind": "import", "name": "paper-t1-003", "file": "table1.txt", "line": 9, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:11 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.200000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/3.a.log", "conductor_log": "data/logs/3.a.conductor.log", "conductor_cputime_seconds": 0.12 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "3.b", "group": [ 3 ], "group_key": "3", "group_bracket": "[3]", "order": 3, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-54", "-318", "-479", "92", "302", "-5", "-54" ], "h": [ "1", "1", "0", "1" ], "f_str": "-54*x^6 - 5*x^5 + 302*x^4 + 92*x^3 - 479*x^2 - 318*x - 54", "h_str": "x^3 + x + 1", "g": [ "-215", "-1270", "-1915", "370", "1210", "-20", "-215" ], "g_str": "-215*x^6 - 20*x^5 + 1210*x^4 + 370*x^3 - 1915*x^2 - 1270*x - 215", "g_scale": "1", "discriminant": "-1658880000000000000000000000", "discriminant_digits": 28, "igusa_clebsch": [ "27817800", "51598476562500", "355152742330078125000", "-1658880000000000000000000000" ], "g2_invariants": [ "-642656500325637801624129/512000000000000", "13988971840727660243/4000000000000", "907932585764603/281250000000" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "-54", "-318", "-479", "92", "302", "-5", "-54" ], "h": [ "1", "1", "0", "1" ] }, "minimal_discriminant": "-405000000000000000000000", "minimal_discriminant_factored": "-1 * 2^21 * 3^4 * 5^22", "conductor": "324", "conductor_factored": "2^2 * 3^4", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "324.a.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-54%2C-318%2C-479%2C92%2C302%2C-5%2C-54%5D%2C%5B1%2C1%2C0%2C1%5D%5D", "snapshot_label": "324.a.2" }, "torsion": { "invariants": [ 3 ], "order": 3, "generators": [ { "order": 3, "mumford": [ "x^2 + x + 1", "25*x + 50", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 21, "point_count_primes": [ 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 3 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 91, "reducible_chi": 34 }, "q_simple": { "certified": true, "prime": 7, "chi": "T^4 + 2*T^3 - 3*T^2 + 14*T + 49" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "cover", "over": "X^3 - 53859375/4*X^2 + 116033291015625/8*X - 138877345184326171875/64", "detail": "a nonconstant map of degree 2 from y^2 = g to the genus-1 curve v^2 = (4)u^3 + (0)u^2 + (-w)u + (1/37405695000*(-1548256*w^2 - 6254107537500*w + 623782079912109375)) over X^3 - 53859375/4*X^2 + 116033291015625/8*X - 138877345184326171875/64, (x,y) -> (p/q, y*hh/q^2) with p = 1/58446398437500*(624*w^2 - 12088437500*w - 5401994326171875)*x^2 + 1/29223199218750*(1312*w^2 - 24895000000*w + 17443671416015625)*x + 1/58446398437500*(-3376*w^2 + 32794375000*w - 31187170107421875), q = x^2 + 1/73057998046875*(-32*w^2 + 440450000*w - 352397402343750)*x + 1/43834798828125*(16*w^2 - 203947500*w + 190810300781250), hh = 1/14611599609375*(-16*w^2 + 301612500*w - 322314697265625)*x + 1/14611599609375*(64*w^2 - 718125000*w + 631736806640625); degree-2 map to an elliptic curve over the field defined by X^3-53859375/4*X^2+116033291015625/8*X-138877345184326171875/64; E: v^2 = 4u^3 - g2 u - g3 with g2 = w, g3 = 1/37405695000*(1548256*w^2+6254107537500*w-623782079912109375) [Genus2Torsion/verify_split_certificates.m, cover 18]" }, "isogenous_product": null, "cover_submitted": true, "cover_accepted": true, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 42, "reducible_chi": 20 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 3 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-54", "-318", "-479", "92", "302", "-5", "-54" ], "h": [ "1", "1", "0", "1" ], "claimed_group": [ 3 ], "claimed_class": "split", "cover": { "field": "X^3-53859375/4*X^2+116033291015625/8*X-138877345184326171875/64", "cremona": "", "cubic": [ "4", "0", "-(w)", "-(1/37405695000*(1548256*w^2+6254107537500*w-623782079912109375))" ], "p": [ "1/58446398437500*(-3376*w^2+32794375000*w-31187170107421875)", "1/29223199218750*(1312*w^2-24895000000*w+17443671416015625)", "1/58446398437500*(624*w^2-12088437500*w-5401994326171875)" ], "q": [ "1/43834798828125*(16*w^2-203947500*w+190810300781250)", "1/73057998046875*(-32*w^2+440450000*w-352397402343750)", "1" ], "hh": [ "1/14611599609375*(64*w^2-718125000*w+631736806640625)", "1/14611599609375*(-16*w^2+301612500*w-322314697265625)" ], "note": "degree-2 map to an elliptic curve over the field defined by X^3-53859375/4*X^2+116033291015625/8*X-138877345184326171875/64; E: v^2 = 4u^3 - g2 u - g3 with g2 = w, g3 = 1/37405695000*(1548256*w^2+6254107537500*w-623782079912109375) [Genus2Torsion/verify_split_certificates.m, cover 18]" }, "lmfdb_label_given": "324.a.2" }, "source": { "kind": "import", "name": "paper-t2-003", "file": "table2.txt", "line": 9, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:33:17 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.21000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/3.b.log", "conductor_log": "data/logs/3.b.conductor.log", "conductor_cputime_seconds": 0.48 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "3.c", "group": [ 3 ], "group_key": "3", "group_bracket": "[3]", "order": 3, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-12", "-15", "9", "31", "9", "-15", "-12" ], "h": [ "1", "1", "1", "1" ], "f_str": "-12*x^6 - 15*x^5 + 9*x^4 + 31*x^3 + 9*x^2 - 15*x - 12", "h_str": "x^3 + x^2 + x + 1", "g": [ "-47", "-58", "39", "128", "39", "-58", "-47" ], "g_str": "-47*x^6 - 58*x^5 + 39*x^4 + 128*x^3 + 39*x^2 - 58*x - 47", "g_scale": "1", "discriminant": "-3469312", "discriminant_digits": 7, "igusa_clebsch": [ "-321632", "6438459712", "-518064193667072", "-3469312" ], "g2_invariants": [ "105037970421355597057024/847", "18267839107785466368/847", "454326923025280/121" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "-12", "-56", "-94", "-83", "-59", "-21", "-7" ], "h": [ "1" ] }, "minimal_discriminant": "-847", "minimal_discriminant_factored": "-1 * 7 * 11^2", "conductor": "847", "conductor_factored": "11 * 77 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 11 * 77 for the elliptic curves ['0', '-1', '1', '-7820', '-263580'] and ['0', '1', '1', '-89', '295'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "847" }, "lmfdb": { "label": "847.d.847.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/847/d/847/1", "snapshot_label": "" }, "torsion": { "invariants": [ 3 ], "order": 3, "generators": [ { "order": 3, "mumford": [ "x^2 + 29/18*x + 1", "11/324*x + 1/18", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "the submitted generators generate a subgroup of order 3, but the point-count bound is 15; TorsionSubgroup was run", "generators_submitted": 1, "point_count_gcd": 15, "point_count_primes": [ 3, 5, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 3 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 91, "reducible_chi": 91 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (z : y : x) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "0", "-1", "1", "-7820", "-263580" ], [ "0", "1", "1", "-89", "295" ] ], "conductors": [ "11", "77" ], "conductor": "847", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 42, "reducible_chi": 42 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "LMFDB 847.d.847.1, https://www.lmfdb.org/Genus2Curve/Q/847/d/847/1", "sources": [], "rm": false, "notes": "Q-split witness: the census has only geometrically simple and Q-simple (geometrically split) curves with this torsion group. Not a new curve. LMFDB: torsion [3], not simple over Q (is_simple_base = false). Split over Q by the involution (x, y) -> (1/x, y/x^3) (f and h are palindromic). Output of pipeline/magma/prepare_submission.m (Magma V2.29-4): J(Q)_tors = [3]; gcd of #J(F_p) over 25 good primes = 15 (not sharp); no strict prime below 500; conductor 847 = 7 * 11^2.", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "-12", "-15", "9", "31", "9", "-15", "-12" ], "h": [ "1", "1", "1", "1" ], "claimed_group": [ 3 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "issue", "number": 4, "url": "https://github.com/Genus-2-torsion/Genus-2-torsion.github.io/issues/4" }, "dates": { "submitted": "2026-10-14", "verified": "2026-10-14 10:49:07 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.46000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/3.c.log", "conductor_log": "data/logs/3.c.conductor.log", "conductor_cputime_seconds": 0.71 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "4.a", "group": [ 4 ], "group_key": "4", "group_bracket": "[4]", "order": 4, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "-1", "0", "1" ], "h": [ "1", "0", "0", "1" ], "f_str": "x^3 - x", "h_str": "x^3 + 1", "g": [ "1", "-4", "0", "6", "0", "0", "1" ], "g_str": "x^6 + 6*x^3 - 4*x + 1", "g_scale": "1", "discriminant": "-8765440", "discriminant_digits": 7, "igusa_clebsch": [ "-24", "13284", "-1135512", "-8765440" ], "g2_invariants": [ "243/2140", "-1863/1070", "-4176/535" ], "rational_weierstrass_points": 1, "minimal_model": { "f": [ "0", "-1", "0", "1" ], "h": [ "1", "0", "0", "1" ] }, "minimal_discriminant": "-2140", "minimal_discriminant_factored": "-1 * 2^2 * 5 * 107", "conductor": "1070", "conductor_factored": "2 * 5 * 107", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "1070.a.2140.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/1070/a/2140/1", "snapshot_label": "" }, "torsion": { "invariants": [ 4 ], "order": 4, "generators": [ { "order": 4, "mumford": [ "x", "x^3 + 1", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 4, "point_count_primes": [ 3, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 4 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 17, "chi": "T^4 + 4*T^3 + 6*T^2 + 68*T + 289", "primes_tried_below": 500, "admissible_primes": 5, "reducible_chi": 4 }, "q_simple": { "certified": true, "prime": 17, "chi": "T^4 + 4*T^3 + 6*T^2 + 68*T + 289" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 43, "reducible_chi": 15 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 4 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "-1", "0", "1" ], "h": [ "1", "0", "0", "1" ], "claimed_group": [ 4 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "1070.a.2140.1" }, "source": { "kind": "import", "name": "paper-t1-004", "file": "table1.txt", "line": 10, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:12 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.260000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/4.a.log", "conductor_log": "data/logs/4.a.conductor.log", "conductor_cputime_seconds": 0.2 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "4.b", "group": [ 4 ], "group_key": "4", "group_bracket": "[4]", "order": 4, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-7536", "-4824", "-4239", "-744", "-304", "76", "-4" ], "h": [ "0", "0", "1", "1" ], "f_str": "-4*x^6 + 76*x^5 - 304*x^4 - 744*x^3 - 4239*x^2 - 4824*x - 7536", "h_str": "x^3 + x^2", "g": [ "-30144", "-19296", "-16956", "-2976", "-1215", "306", "-15" ], "g_str": "-15*x^6 + 306*x^5 - 1215*x^4 - 2976*x^3 - 16956*x^2 - 19296*x - 30144", "g_scale": "1", "discriminant": "-3544912911056091616283408933584896", "discriminant_digits": 34, "igusa_clebsch": [ "-621186624", "22912741487578176", "-3589794001362244677760512", "-3544912911056091616283408933584896" ], "g2_invariants": [ "48026592430337743623160621144/14725391615068258683", "1199128756610256996174902945/176704699380819104196", "-1687781526798200476039424/397585573606842984441" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "-7536", "-4824", "-4239", "-744", "-304", "76", "-4" ], "h": [ "0", "0", "1", "1" ] }, "minimal_discriminant": "-865457253675803617256691634176", "minimal_discriminant_factored": "-1 * 2^12 * 3^22 * 13^12 * 17^2", "conductor": "289", "conductor_factored": "17 * 17 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 17 * 17 for the elliptic curves ['1', '-1', '1', '-1', '0'] and ['1', '-1', '1', '-91', '-310'] to whose product J is isogenous over Q", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "" }, "lmfdb": { "label": "289.a.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-7536%2C-4824%2C-4239%2C-744%2C-304%2C76%2C-4%5D%2C%5B0%2C0%2C1%2C1%5D%5D", "snapshot_label": "289.a.2" }, "torsion": { "invariants": [ 4 ], "order": 4, "generators": [ { "order": 4, "mumford": [ "x^2 + x + 13/2", "-99/2*x - 108", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 16, "point_count_primes": [ 7, 11, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113 ], "claimed": [ 4 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 90, "reducible_chi": 90 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x + 24/11*z : 312/1331*x^3 - 3120/1331*x^2*z - 7176/1331*x*z^2 + 2197/1331*y - 3744/1331*z^3 : 2/11*x - z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "-1", "1", "-1", "0" ], [ "1", "-1", "1", "-91", "-310" ] ], "conductors": [ "17", "17" ], "conductor": "289", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 41, "reducible_chi": 41 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 4 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-7536", "-4824", "-4239", "-744", "-304", "76", "-4" ], "h": [ "0", "0", "1", "1" ], "claimed_group": [ 4 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "289.a.2" }, "source": { "kind": "import", "name": "paper-t2-004", "file": "table2.txt", "line": 10, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:35:49 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.24000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/4.b.log", "conductor_log": "data/logs/4.b.conductor.log", "conductor_cputime_seconds": null }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "4.c", "group": [ 4 ], "group_key": "4", "group_bracket": "[4]", "order": 4, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-9", "-44", "11", "80", "105", "56", "16" ], "h": [ "0", "1" ], "f_str": "16*x^6 + 56*x^5 + 105*x^4 + 80*x^3 + 11*x^2 - 44*x - 9", "h_str": "x", "g": [ "-36", "-176", "45", "320", "420", "224", "64" ], "g_str": "64*x^6 + 224*x^5 + 420*x^4 + 320*x^3 + 45*x^2 - 176*x - 36", "g_scale": "1", "discriminant": "15872000000000000000", "discriminant_digits": 20, "igusa_clebsch": [ "-712000", "73249000000", "-10758419000000000", "15872000000000000000" ], "g2_invariants": [ "-44672475592/31", "9767345495/124", "-3714949" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "-9", "-44", "11", "80", "105", "56", "16" ], "h": [ "0", "1" ] }, "minimal_discriminant": "3875000000000000", "minimal_discriminant_factored": "2^12 * 5^15 * 31", "conductor": "775", "conductor_factored": "5^2 * 31", "conductor_note": "taken from the LMFDB label 775.a.1: Magma's genus-2 Conductor gave 793600, with the exponent of 2 from Ogg's formula (no correctness guarantee when v_2(disc) >= 12); the odd parts agree", "conductor_source": "lmfdb label", "conductor_magma": "793600" }, "lmfdb": { "label": "775.a.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-9%2C-44%2C11%2C80%2C105%2C56%2C16%5D%2C%5B0%2C1%5D%5D", "snapshot_label": "775.a.1" }, "torsion": { "invariants": [ 4 ], "order": 4, "generators": [ { "order": 4, "mumford": [ "1", "8*x^3 + 14*x^2", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "the submitted generators generate a subgroup of order 4, but the point-count bound is 8; TorsionSubgroup was run", "generators_submitted": 1, "point_count_gcd": 8, "point_count_primes": [ 3, 7, 11, 13, 17, 19, 23, 29, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 4 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 50 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 - 2*T^2 + 9" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "richelot", "over": "Q", "detail": "(2,2)-isogenous over Q to a Weil restriction of an elliptic curve E over Q(sqrt(5)) (the surface is E x E^sigma)" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 24 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": 2026, "historical": false }, "reference": "LMFDB extended genus 2 database (Booker–Sutherland), curve 775.a.1 (table g2c_curves_new, label 775.a1 on devmirror.lmfdb.xyz, read 2026-10-16)", "sources": [ "BookerSutherland" ], "rm": false, "notes": "First Q-simple, geometrically split example of [4] for the census (the gsplit cell is empty). Found by querying the LMFDB genus 2 data (devmirror.lmfdb.xyz, g2c_curves_new: is_simple_base = true, is_simple_geom = false, torsion_subgroup = [4]; geom_end_alg = Q x Q, end_alg = Q; smallest conductor among the curves of this group that the census verifier certifies). LMFDB conductor 775. Checked locally with the census's own pipeline/magma/verify_lib.m (Magma V2.30-1): J(Q)_tors = [4], class gsplit, Q-simplicity prime 3 (chi = T^4 - 2*T^2 + 9), splitness certificate: richelot over Q ((2,2)-isogenous over Q to a Weil restriction of an elliptic curve E over Q(sqrt(5)) (the surface is E x E^sigma)). Note: Magma's genus-2 Conductor on the reduced minimal model gives 793600 (2-part by Ogg's formula), not the LMFDB value 775. Submitted by F. Najman; the curve is due to the database cited under Reference. Rigorous conductor: J is (2,2)-isogenous over Q to Res_(K/Q) E with K the quadratic field of discriminant 5, so N(J) = |d_K|^2 * Norm(f_E) = 25 * 31 = 775 (computed with Magma; conductor is an isogeny invariant).", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "-9", "-44", "11", "80", "105", "56", "16" ], "h": [ "0", "1" ], "claimed_group": [ 4 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "775.a.1" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_4.json" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-17 06:28:59 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "0.870000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/4.c.log", "conductor_log": "data/logs/4.c.conductor.log", "conductor_cputime_seconds": 1.15 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "5.a", "group": [ 5 ], "group_key": "5", "group_bracket": "[5]", "order": 5, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "-6", "11", "-19", "14", "-9", "1" ], "h": [ "1" ], "f_str": "x^5 - 9*x^4 + 14*x^3 - 19*x^2 + 11*x - 6", "h_str": "1", "g": [ "-23", "44", "-76", "56", "-36", "4" ], "g_str": "4*x^5 - 36*x^4 + 56*x^3 - 76*x^2 + 44*x - 23", "g_scale": "1", "discriminant": "70912", "discriminant_digits": 5, "igusa_clebsch": [ "-17920", "21928192", "-96756463616", "1134592" ], "g2_invariants": [ "-56394933862400000/277", "217505333248000/277", "-824813158400/277" ], "rational_weierstrass_points": 1, "minimal_model": { "f": [ "-6", "11", "-19", "14", "-9", "1" ], "h": [ "1" ] }, "minimal_discriminant": "277", "minimal_discriminant_factored": "277", "conductor": "277", "conductor_factored": "277", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "277.a.277.2", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/277/a/277/2", "snapshot_label": "" }, "torsion": { "invariants": [ 5 ], "order": 5, "generators": [ { "order": 5, "mumford": [ "x^2 - 7/9*x + 8/9", "2/27*x - 1/27", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 15, "point_count_primes": [ 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101 ], "claimed": [ 5 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 3, "chi": "T^4 + T^3 + T^2 + 3*T + 9", "primes_tried_below": 500, "admissible_primes": 1, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 + T^3 + T^2 + 3*T + 9" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 45, "reducible_chi": 14 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 5 (source: \\cite{Ogg1973})", "sources": [ "Ogg1973" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-6", "11", "-19", "14", "-9", "1" ], "h": [ "1" ], "claimed_group": [ 5 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "277.a.277.2" }, "source": { "kind": "import", "name": "paper-t1-005", "file": "table1.txt", "line": 11, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:12 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.250000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/5.a.log", "conductor_log": "data/logs/5.a.conductor.log", "conductor_cputime_seconds": 0.11 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "5.b", "group": [ 5 ], "group_key": "5", "group_bracket": "[5]", "order": 5, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-344", "348", "-334", "17", "14", "-36", "-8" ], "h": [ "0" ], "f_str": "-8*x^6 - 36*x^5 + 14*x^4 + 17*x^3 - 334*x^2 + 348*x - 344", "h_str": "0", "g": [ "-1376", "1392", "-1336", "68", "56", "-144", "-32" ], "g_str": "-32*x^6 - 144*x^5 + 56*x^4 + 68*x^3 - 1336*x^2 + 1392*x - 1376", "g_scale": "1", "discriminant": "115091087670455100768256", "discriminant_digits": 24, "igusa_clebsch": [ "-17360800", "19855296692224", "-85439612111571386368", "115091087670455100768256" ], "g2_invariants": [ "-47000099587093154379892578125/27439853589643264", "846620985909327133017140625/219518828717146112", "1560040714626774135625/439037657434292224" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "-344", "348", "-334", "17", "14", "-36", "-8" ], "h": [] }, "minimal_discriminant": "28098410075794702336", "minimal_discriminant_factored": "2^24 * 7^12 * 11^2", "conductor": "121", "conductor_factored": "11 * 11 (product of the elliptic conductors)", "conductor_note": "Magma's genus-2 Conductor gave 126877696 (Ogg's formula at 2, no correctness guarantee); the rigorous value is N(E1) N(E2) = 11 * 11 for the elliptic curves ['0', '-1', '1', '0', '0'] and ['0', '-1', '1', '-7820', '-263580'] to whose product J is isogenous over Q", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "126877696" }, "lmfdb": { "label": "121.a.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-344%2C348%2C-334%2C17%2C14%2C-36%2C-8%5D%2C%5B0%5D%5D", "snapshot_label": "121.a.2" }, "torsion": { "invariants": [ 5 ], "order": 5, "generators": [ { "order": 5, "mumford": [ "x^2 - 3/2*x + 5/3", "-11/3*x + 6", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 25, "point_count_primes": [ 3, 5, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 5 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x - 11/4*z : 343/64*y : -3/4*x - z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "0", "-1", "1", "0", "0" ], [ "0", "-1", "1", "-7820", "-263580" ] ], "conductors": [ "11", "11" ], "conductor": "121", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 5 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-344", "348", "-334", "17", "14", "-36", "-8" ], "h": [ "0" ], "claimed_group": [ 5 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "121.a.2" }, "source": { "kind": "import", "name": "paper-t2-005", "file": "table2.txt", "line": 11, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:35:50 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.980000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/5.b.log", "conductor_log": "data/logs/5.b.conductor.log", "conductor_cputime_seconds": 0.25 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "5.c", "group": [ 5 ], "group_key": "5", "group_bracket": "[5]", "order": 5, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "357", "30", "86", "-21", "2", "-3" ], "h": [ "1", "0", "0", "1" ], "f_str": "-3*x^5 + 2*x^4 - 21*x^3 + 86*x^2 + 30*x + 357", "h_str": "x^3 + 1", "g": [ "1429", "120", "344", "-82", "8", "-12", "1" ], "g_str": "x^6 - 12*x^5 + 8*x^4 - 82*x^3 + 344*x^2 + 120*x + 1429", "g_scale": "1", "discriminant": "838629671142862962688", "discriminant_digits": 21, "igusa_clebsch": [ "-404248", "25504858756", "-2179521547133944", "838629671142862962688" ], "g2_invariants": [ "-6436343/4", "401511/4", "-12167/4" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "357", "30", "86", "-21", "2", "-3" ], "h": [ "1", "0", "0", "1" ] }, "minimal_discriminant": "204743572056363028", "minimal_discriminant_factored": "2^2 * 13^15", "conductor": "676", "conductor_factored": "2^2 * 13^2", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "676.c.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B357%2C30%2C86%2C-21%2C2%2C-3%5D%2C%5B1%2C0%2C0%2C1%5D%5D", "snapshot_label": "676.c.1" }, "torsion": { "invariants": [ 5 ], "order": 5, "generators": [ { "order": 5, "mumford": [ "x^2 + 5/2*x + 21/2", "13/4*x + 143/4", "2" ] } ], "method": "generators", "generators_note": "", "generators_submitted": 1, "point_count_gcd": 5, "point_count_primes": [ 3, 5, 7, 11, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 5 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 93, "reducible_chi": 48 }, "q_simple": { "certified": true, "prime": 5, "chi": "T^4 - T^2 + 25" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "richelot", "over": "Q", "detail": "(2,2)-isogenous over Q to a Weil restriction of an elliptic curve E over Q(sqrt(13)) (the surface is E x E^sigma)" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 44, "reducible_chi": 25 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": 2026, "historical": false }, "reference": "LMFDB extended genus 2 database (Booker–Sutherland), curve 676.c.1 (table g2c_curves_new, label 676.c1 on devmirror.lmfdb.xyz, read 2026-10-16)", "sources": [ "BookerSutherland" ], "rm": false, "notes": "First Q-simple, geometrically split example of [5] for the census (the gsplit cell is empty). Found by querying the LMFDB genus 2 data (devmirror.lmfdb.xyz, g2c_curves_new: is_simple_base = true, is_simple_geom = false, torsion_subgroup = [5]; geom_end_alg = M_2(Q), end_alg = CM; smallest conductor among the curves of this group that the census verifier certifies). LMFDB conductor 676. Checked locally with the census's own pipeline/magma/verify_lib.m (Magma V2.30-1): J(Q)_tors = [5], class gsplit, Q-simplicity prime 5 (chi = T^4 - T^2 + 25), splitness certificate: richelot over Q ((2,2)-isogenous over Q to a Weil restriction of an elliptic curve E over Q(sqrt(13)) (the surface is E x E^sigma)). Submitted by F. Najman; the curve is due to the database cited under Reference. Rigorous conductor: J is (2,2)-isogenous over Q to Res_(K/Q) E with K the quadratic field of discriminant 13, so N(J) = |d_K|^2 * Norm(f_E) = 169 * 4 = 676 (computed with Magma; conductor is an isogeny invariant).", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "357", "30", "86", "-21", "2", "-3" ], "h": [ "1", "0", "0", "1" ], "claimed_group": [ 5 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "676.c.1" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_5.json" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-17 06:29:00 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "0.800000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/5.c.log", "conductor_log": "data/logs/5.c.conductor.log", "conductor_cputime_seconds": 0.24 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "6.a", "group": [ 6 ], "group_key": "6", "group_bracket": "[6]", "order": 6, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "3", "21", "46", "26", "-12", "1" ], "h": [ "0", "1", "1" ], "f_str": "x^5 - 12*x^4 + 26*x^3 + 46*x^2 + 21*x + 3", "h_str": "x^2 + x", "g": [ "12", "84", "185", "106", "-47", "4" ], "g_str": "4*x^5 - 47*x^4 + 106*x^3 + 185*x^2 + 84*x + 12", "g_scale": "1", "discriminant": "265728", "discriminant_digits": 6, "igusa_clebsch": [ "219976", "1339396", "98660530696", "4251648" ], "g2_invariants": [ "15719059879327073637257/1038", "109111794064913809345/173", "18168889743107727596/519" ], "rational_weierstrass_points": 1, "minimal_model": { "f": [ "0", "-1", "-12", "-27", "46", "-21", "3" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "1038", "minimal_discriminant_factored": "2 * 3 * 173", "conductor": "1038", "conductor_factored": "2 * 3 * 173", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "1038.a.1038.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/1038/a/1038/1", "snapshot_label": "" }, "torsion": { "invariants": [ 6 ], "order": 6, "generators": [ { "order": 6, "mumford": [ "x^2 - 4*x - 2", "-9*x - 4", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 12, "point_count_primes": [ 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 6 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 17, "chi": "T^4 - 4*T^3 + 10*T^2 - 68*T + 289", "primes_tried_below": 500, "admissible_primes": 5, "reducible_chi": 2 }, "q_simple": { "certified": true, "prime": 5, "chi": "T^4 + 3*T^3 + 4*T^2 + 15*T + 25" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 43, "reducible_chi": 13 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 6 (source: \\cite{Flynn1991})", "sources": [ "Flynn1991" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "3", "21", "46", "26", "-12", "1" ], "h": [ "0", "1", "1" ], "claimed_group": [ 6 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "1038.a.1038.1" }, "source": { "kind": "import", "name": "paper-t1-006", "file": "table1.txt", "line": 12, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:13 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.280000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/6.a.log", "conductor_log": "data/logs/6.a.conductor.log", "conductor_cputime_seconds": 0.35 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "6.b", "group": [ 6 ], "group_key": "6", "group_bracket": "[6]", "order": 6, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-92", "42", "3", "37", "-12", "-3", "-5" ], "h": [ "0", "1", "1" ], "f_str": "-5*x^6 - 3*x^5 - 12*x^4 + 37*x^3 + 3*x^2 + 42*x - 92", "h_str": "x^2 + x", "g": [ "-368", "168", "13", "150", "-47", "-12", "-20" ], "g_str": "-20*x^6 - 12*x^5 - 47*x^4 + 150*x^3 + 13*x^2 + 168*x - 368", "g_scale": "1", "discriminant": "-3136000000000000", "discriminant_digits": 16, "igusa_clebsch": [ "-1702264", "165439562500", "-71443609436875000", "-3136000000000000" ], "g2_invariants": [ "436198972565920927085724143/765625000000", "393062363938781788055883/191406250000", "1831208036823726634/244140625" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "-92", "42", "3", "37", "-12", "-3", "-5" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "-765625000000", "minimal_discriminant_factored": "-1 * 2^6 * 5^12 * 7^2", "conductor": "196", "conductor_factored": "14 * 14 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 14 * 14 for the elliptic curves ['1', '0', '1', '-2731', '-55146'] and ['1', '0', '1', '-11', '12'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "196" }, "lmfdb": { "label": "196.a.3", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-92%2C42%2C3%2C37%2C-12%2C-3%2C-5%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "196.a.3" }, "torsion": { "invariants": [ 6 ], "order": 6, "generators": [ { "order": 6, "mumford": [ "x^2 + x + 3", "4*x - 7", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 36, "point_count_primes": [ 3, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 6 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x - 8*z : -3*x^3 + 72*x^2*z + 174*x*z^2 + 125*y + 36*z^3 : -3*x - z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "0", "1", "-2731", "-55146" ], [ "1", "0", "1", "-11", "12" ] ], "conductors": [ "14", "14" ], "conductor": "196", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 6 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-92", "42", "3", "37", "-12", "-3", "-5" ], "h": [ "0", "1", "1" ], "claimed_group": [ 6 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "196.a.3" }, "source": { "kind": "import", "name": "paper-t2-006", "file": "table2.txt", "line": 12, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:35:52 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.23000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/6.b.log", "conductor_log": "data/logs/6.b.conductor.log", "conductor_cputime_seconds": 0.23 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "6.c", "group": [ 6 ], "group_key": "6", "group_bracket": "[6]", "order": 6, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-7", "27", "-18", "-23", "6", "3" ], "h": [ "1", "0", "0", "1" ], "f_str": "3*x^5 + 6*x^4 - 23*x^3 - 18*x^2 + 27*x - 7", "h_str": "x^3 + 1", "g": [ "-27", "108", "-72", "-90", "24", "12", "1" ], "g_str": "x^6 + 12*x^5 + 24*x^4 - 90*x^3 - 72*x^2 + 108*x - 27", "g_scale": "1", "discriminant": "4290437984256", "discriminant_digits": 13, "igusa_clebsch": [ "134568", "186790212", "9223002035784", "4290437984256" ], "g2_invariants": [ "93851287159343/73", "3265084367601/73", "1106555779" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "-7", "27", "-18", "-23", "6", "3" ], "h": [ "1", "0", "0", "1" ] }, "minimal_discriminant": "1047470211", "minimal_discriminant_factored": "3^15 * 73", "conductor": "657", "conductor_factored": "3^2 * 73", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "657.a.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-7%2C27%2C-18%2C-23%2C6%2C3%5D%2C%5B1%2C0%2C0%2C1%5D%5D", "snapshot_label": "657.a.1" }, "torsion": { "invariants": [ 6 ], "order": 6, "generators": [ { "order": 6, "mumford": [ "x^2 + 3*x - 3/2", "21/2*x - 9/2", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "the submitted generators generate a subgroup of order 6, but the point-count bound is 12; TorsionSubgroup was run", "generators_submitted": 1, "point_count_gcd": 12, "point_count_primes": [ 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 6 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 49 }, "q_simple": { "certified": true, "prime": 5, "chi": "T^4 - 2*T^2 + 25" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "richelot", "over": "Q", "detail": "(2,2)-isogenous over Q to a Weil restriction of an elliptic curve E over Q(sqrt(-3)) (the surface is E x E^sigma)" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 22 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": 2026, "historical": false }, "reference": "LMFDB extended genus 2 database (Booker–Sutherland), curve 657.a.1 (table g2c_curves_new, label 657.a1 on devmirror.lmfdb.xyz, read 2026-10-16)", "sources": [ "BookerSutherland" ], "rm": false, "notes": "First Q-simple, geometrically split example of [6] for the census (the gsplit cell is empty). Found by querying the LMFDB genus 2 data (devmirror.lmfdb.xyz, g2c_curves_new: is_simple_base = true, is_simple_geom = false, torsion_subgroup = [6]; geom_end_alg = Q x Q, end_alg = Q; smallest conductor among the curves of this group that the census verifier certifies). LMFDB conductor 657. Checked locally with the census's own pipeline/magma/verify_lib.m (Magma V2.30-1): J(Q)_tors = [6], class gsplit, Q-simplicity prime 5 (chi = T^4 - 2*T^2 + 25), splitness certificate: richelot over Q ((2,2)-isogenous over Q to a Weil restriction of an elliptic curve E over Q(sqrt(-3)) (the surface is E x E^sigma)). Submitted by F. Najman; the curve is due to the database cited under Reference. Rigorous conductor: J is (2,2)-isogenous over Q to Res_(K/Q) E with K the quadratic field of discriminant -3, so N(J) = |d_K|^2 * Norm(f_E) = 9 * 73 = 657 (computed with Magma; conductor is an isogeny invariant).", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "-7", "27", "-18", "-23", "6", "3" ], "h": [ "1", "0", "0", "1" ], "claimed_group": [ 6 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "657.a.1" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_6.json" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-17 06:29:01 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "0.860000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/6.c.log", "conductor_log": "data/logs/6.c.conductor.log", "conductor_cputime_seconds": 0.2 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "7.a", "group": [ 7 ], "group_key": "7", "group_bracket": "[7]", "order": 7, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "-2", "3", "0", "-3", "0", "1" ], "h": [ "0", "0", "0", "1" ], "f_str": "x^5 - 3*x^3 + 3*x - 2", "h_str": "x^3", "g": [ "-8", "12", "0", "-12", "0", "4", "1" ], "g_str": "x^6 + 4*x^5 - 12*x^3 + 12*x - 8", "g_scale": "1", "discriminant": "1888256", "discriminant_digits": 7, "igusa_clebsch": [ "4704", "2304", "4253184", "1888256" ], "g2_invariants": [ "70288881159168/461", "2923824242304/461", "161508086208/461" ], "rational_weierstrass_points": 1, "minimal_model": { "f": [ "-3", "-3", "2", "1", "-2", "-1" ], "h": [ "1", "1", "1", "1" ] }, "minimal_discriminant": "461", "minimal_discriminant_factored": "461", "conductor": "461", "conductor_factored": "461", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "461.a.461.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/461/a/461/1", "snapshot_label": "" }, "torsion": { "invariants": [ 7 ], "order": 7, "generators": [ { "order": 7, "mumford": [ "x^2 + x - 3", "1", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 7, "point_count_primes": [ 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101 ], "claimed": [ 7 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 5, "chi": "T^4 - T^3 + T^2 - 5*T + 25", "primes_tried_below": 500, "admissible_primes": 2, "reducible_chi": 1 }, "q_simple": { "certified": true, "prime": 5, "chi": "T^4 - T^3 + T^2 - 5*T + 25" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 45, "reducible_chi": 7 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 7 (source: \\cite{Ogg1973})", "sources": [ "Ogg1973" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-2", "3", "0", "-3", "0", "1" ], "h": [ "0", "0", "0", "1" ], "claimed_group": [ 7 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "461.a.461.1" }, "source": { "kind": "import", "name": "paper-t1-007", "file": "table1.txt", "line": 13, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:14 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.250000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/7.a.log", "conductor_log": "data/logs/7.a.conductor.log", "conductor_cputime_seconds": 0.12 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "7.b", "group": [ 7 ], "group_key": "7", "group_bracket": "[7]", "order": 7, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-3", "-4", "2", "7", "2", "-4", "-3" ], "h": [ "1", "0", "0", "1" ], "f_str": "-3*x^6 - 4*x^5 + 2*x^4 + 7*x^3 + 2*x^2 - 4*x - 3", "h_str": "x^3 + 1", "g": [ "-11", "-16", "8", "30", "8", "-16", "-11" ], "g_str": "-11*x^6 - 16*x^5 + 8*x^4 + 30*x^3 + 8*x^2 - 16*x - 11", "g_scale": "1", "discriminant": "-22682796032", "discriminant_digits": 11, "igusa_clebsch": [ "-14424", "12978852", "-46804610136", "-22682796032" ], "g2_invariants": [ "19053669755529243/5537792", "744373495629/2768896", "16627888035/1384448" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "-3", "12", "-20", "14", "-10", "3", "-1" ], "h": [ "1", "1", "1" ] }, "minimal_discriminant": "-5537792", "minimal_discriminant_factored": "-1 * 2^15 * 13^2", "conductor": "676", "conductor_factored": "26 * 26 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 26 * 26 for the elliptic curves ['1', '0', '1', '-460', '-3830'] and ['1', '-1', '1', '-3', '3'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "676" }, "lmfdb": { "label": "676.b.3", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-3%2C-4%2C2%2C7%2C2%2C-4%2C-3%5D%2C%5B1%2C0%2C0%2C1%5D%5D", "snapshot_label": "676.b.3" }, "torsion": { "invariants": [ 7 ], "order": 7, "generators": [ { "order": 7, "mumford": [ "x^2 + x + 1", "4", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 21, "point_count_primes": [ 3, 5, 7, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 7 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (z : y : x) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "0", "1", "-460", "-3830" ], [ "1", "-1", "1", "-3", "3" ] ], "conductors": [ "26", "26" ], "conductor": "676", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 7 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-3", "-4", "2", "7", "2", "-4", "-3" ], "h": [ "1", "0", "0", "1" ], "claimed_group": [ 7 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "676.b.3" }, "source": { "kind": "import", "name": "paper-t2-007", "file": "table2.txt", "line": 13, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:35:53 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.03000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/7.b.log", "conductor_log": "data/logs/7.b.conductor.log", "conductor_cputime_seconds": 0.19 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "7.c", "group": [ 7 ], "group_key": "7", "group_bracket": "[7]", "order": 7, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "2", "2", "-8", "-6", "12", "10", "2" ], "h": [ "0" ], "f_str": "2*x^6 + 10*x^5 + 12*x^4 - 6*x^3 - 8*x^2 + 2*x + 2", "h_str": "0", "g": [ "8", "8", "-32", "-24", "48", "40", "8" ], "g_str": "8*x^6 + 40*x^5 + 48*x^4 - 24*x^3 - 32*x^2 + 8*x + 8", "g_scale": "1", "discriminant": "-78391743086592", "discriminant_digits": 14, "igusa_clebsch": [ "25472", "51970048", "319658393600", "-78391743086592" ], "g2_invariants": [ "-312079600999/18252", "3254687387/16224", "8276609/10816" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "2", "-10", "12", "6", "-8", "-2", "2" ], "h": [] }, "minimal_discriminant": "-19138609152", "minimal_discriminant_factored": "-1 * 2^22 * 3^3 * 13^2", "conductor": "1521", "conductor_factored": "3^2 * 13^2", "conductor_note": "taken from the LMFDB label 1521.d.1 (Magma's genus-2 Conductor failed on this curve: Fibre blowups over number fields not yet implemented)", "conductor_source": "lmfdb label" }, "lmfdb": { "label": "1521.d.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B2%2C2%2C-8%2C-6%2C12%2C10%2C2%5D%2C%5B0%5D%5D", "snapshot_label": "1521.d.1" }, "torsion": { "invariants": [ 7 ], "order": 7, "generators": [ { "order": 7, "mumford": [ "x^2 + 3*x + 1", "2*x", "2" ] } ], "method": "generators", "generators_note": "", "generators_submitted": 1, "point_count_gcd": 7, "point_count_primes": [ 5, 7, 11, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 7 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 26 }, "q_simple": { "certified": true, "prime": 5, "chi": "T^4 + 2*T^2 + 25" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "geometric_involution", "over": "Qbar", "detail": "the geometric automorphism group has order 12 (7 involutions), so the curve is bielliptic over Qbar" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 14 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": 2026, "historical": false }, "reference": "LMFDB extended genus 2 database (Booker–Sutherland), curve 1521.d.1 (table g2c_curves_new, label 1521.d1 on devmirror.lmfdb.xyz, read 2026-10-16)", "sources": [ "BookerSutherland" ], "rm": false, "notes": "First Q-simple, geometrically split example of [7] for the census (the gsplit cell is empty). Found by querying the LMFDB genus 2 data (devmirror.lmfdb.xyz, g2c_curves_new: is_simple_base = true, is_simple_geom = false, torsion_subgroup = [7]; geom_end_alg = M_2(Q), end_alg = CM; smallest conductor among the curves of this group that the census verifier certifies). LMFDB conductor 1521. Checked locally with the census's own pipeline/magma/verify_lib.m (Magma V2.30-1): J(Q)_tors = [7], class gsplit, Q-simplicity prime 5 (chi = T^4 + 2*T^2 + 25), splitness certificate: geometric_involution over Qbar (the geometric automorphism group has order 12 (7 involutions), so the curve is bielliptic over Qbar). Submitted by F. Najman; the curve is due to the database cited under Reference.", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "2", "2", "-8", "-6", "12", "10", "2" ], "h": [ "0" ], "claimed_group": [ 7 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "1521.d.1" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_7.json" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-17 06:29:02 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "0.790000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/7.c.log", "conductor_log": "data/logs/7.c.conductor.log", "conductor_cputime_seconds": null }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "8.a", "group": [ 8 ], "group_key": "8", "group_bracket": "[8]", "order": 8, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "0", "0", "-1", "-2", "-2", "-1" ], "h": [ "1", "1" ], "f_str": "-x^6 - 2*x^5 - 2*x^4 - x^3", "h_str": "x + 1", "g": [ "1", "2", "1", "-4", "-8", "-8", "-4" ], "g_str": "-4*x^6 - 8*x^5 - 8*x^4 - 4*x^3 + x^2 + 2*x + 1", "g_scale": "1", "discriminant": "1900544", "discriminant_digits": 7, "igusa_clebsch": [ "544", "4480", "963840", "1900544" ], "g2_invariants": [ "90870848/29", "2869192/29", "-18496/29" ], "rational_weierstrass_points": 1, "minimal_model": { "f": [ "0", "1", "1", "2", "1", "1" ], "h": [ "0", "1", "0", "1" ] }, "minimal_discriminant": "464", "minimal_discriminant_factored": "2^4 * 29", "conductor": "464", "conductor_factored": "2^4 * 29", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "464.a.464.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/464/a/464/1", "snapshot_label": "" }, "torsion": { "invariants": [ 8 ], "order": 8, "generators": [ { "order": 8, "mumford": [ "x^2 + x", "x + 1", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 16, "point_count_primes": [ 3, 5, 7, 11, 13, 17, 19, 23, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 8 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 3, "chi": "T^4 + T^3 + 2*T^2 + 3*T + 9", "primes_tried_below": 500, "admissible_primes": 1, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 + T^3 + 2*T^2 + 3*T + 9" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 44, "reducible_chi": 14 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 8 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "0", "0", "-1", "-2", "-2", "-1" ], "h": [ "1", "1" ], "claimed_group": [ 8 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "464.a.464.1" }, "source": { "kind": "import", "name": "paper-t1-008", "file": "table1.txt", "line": 14, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:14 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.220000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/8.a.log", "conductor_log": "data/logs/8.a.conductor.log", "conductor_cputime_seconds": 0.13 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "8.b", "group": [ 8 ], "group_key": "8", "group_bracket": "[8]", "order": 8, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-15", "0", "7", "0", "-4" ], "h": [ "0", "1", "0", "1" ], "f_str": "-4*x^4 + 7*x^2 - 15", "h_str": "x^3 + x", "g": [ "-60", "0", "29", "0", "-14", "0", "1" ], "g_str": "x^6 - 14*x^4 + 29*x^2 - 60", "g_scale": "1", "discriminant": "240000000000000", "discriminant_digits": 15, "igusa_clebsch": [ "20896", "3287296", "6472442624", "240000000000000" ], "g2_invariants": [ "7598815157663552/3662109375", "55695698081848/732421875", "311899421504/48828125" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "-15", "0", "7", "0", "-4" ], "h": [ "0", "1", "0", "1" ] }, "minimal_discriminant": "58593750000", "minimal_discriminant_factored": "2^4 * 3 * 5^13", "conductor": "300", "conductor_factored": "15 * 20 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 15 * 20 for the elliptic curves ['1', '1', '1', '35', '-28'] and ['0', '1', '0', '-36', '-140'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "300" }, "lmfdb": { "label": "300.a.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-15%2C0%2C7%2C0%2C-4%5D%2C%5B0%2C1%2C0%2C1%5D%5D", "snapshot_label": "300.a.2" }, "torsion": { "invariants": [ 8 ], "order": 8, "generators": [ { "order": 8, "mumford": [ "1", "x^3", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 48, "point_count_primes": [ 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 8 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x : y : -z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "1", "1", "35", "-28" ], [ "0", "1", "0", "-36", "-140" ] ], "conductors": [ "15", "20" ], "conductor": "300", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 8 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-15", "0", "7", "0", "-4" ], "h": [ "0", "1", "0", "1" ], "claimed_group": [ 8 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "300.a.2" }, "source": { "kind": "import", "name": "paper-t2-008", "file": "table2.txt", "line": 14, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:35:54 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.11000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/8.b.log", "conductor_log": "data/logs/8.b.conductor.log", "conductor_cputime_seconds": 0.23 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "8.c", "group": [ 8 ], "group_key": "8", "group_bracket": "[8]", "order": 8, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-1", "-2", "1", "4", "-1", "-2", "1" ], "h": [ "0" ], "f_str": "x^6 - 2*x^5 - x^4 + 4*x^3 + x^2 - 2*x - 1", "h_str": "0", "g": [ "-4", "-8", "4", "16", "-4", "-8", "4" ], "g_str": "4*x^6 - 8*x^5 - 4*x^4 + 16*x^3 + 4*x^2 - 8*x - 4", "g_scale": "1", "discriminant": "21474836480", "discriminant_digits": 11, "igusa_clebsch": [ "8192", "-32768", "7340032", "21474836480" ], "g2_invariants": [ "1073741824/5", "45088768/5", "475136" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "-1", "-2", "1", "4", "-1", "-2", "1" ], "h": [] }, "minimal_discriminant": "5242880", "minimal_discriminant_factored": "2^20 * 5", "conductor": "2560", "conductor_factored": "2^9 * 5", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "2560.h.3", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-1%2C-2%2C1%2C4%2C-1%2C-2%2C1%5D%2C%5B0%5D%5D", "snapshot_label": "2560.h.3" }, "torsion": { "invariants": [ 8 ], "order": 8, "generators": [ { "order": 8, "mumford": [ "x + 1", "x^3 + 1", "2" ] } ], "method": "generators", "generators_note": "", "generators_submitted": 1, "point_count_gcd": 8, "point_count_primes": [ 3, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 8 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 93, "reducible_chi": 47 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 - 2*T^2 + 9" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "richelot", "over": "Q", "detail": "(2,2)-isogenous over Q to a Weil restriction of an elliptic curve E over Q(sqrt(-1)) (the surface is E x E^sigma)" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 44, "reducible_chi": 23 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": 2026, "historical": false }, "reference": "LMFDB extended genus 2 database (Booker–Sutherland), curve 2560.h.3 (table g2c_curves_new, label 2560.h3 on devmirror.lmfdb.xyz, read 2026-10-16)", "sources": [ "BookerSutherland" ], "rm": false, "notes": "First Q-simple, geometrically split example of [8] for the census (the gsplit cell is empty). Found by querying the LMFDB genus 2 data (devmirror.lmfdb.xyz, g2c_curves_new: is_simple_base = true, is_simple_geom = false, torsion_subgroup = [8]; geom_end_alg = Q x Q, end_alg = Q; smallest conductor among the curves of this group that the census verifier certifies). LMFDB conductor 2560. Checked locally with the census's own pipeline/magma/verify_lib.m (Magma V2.30-1): J(Q)_tors = [8], class gsplit, Q-simplicity prime 3 (chi = T^4 - 2*T^2 + 9), splitness certificate: richelot over Q ((2,2)-isogenous over Q to a Weil restriction of an elliptic curve E over Q(sqrt(-1)) (the surface is E x E^sigma)). Submitted by F. Najman; the curve is due to the database cited under Reference. Rigorous conductor: J is (2,2)-isogenous over Q to Res_(K/Q) E with K the quadratic field of discriminant -4, so N(J) = |d_K|^2 * Norm(f_E) = 16 * 160 = 2560 (computed with Magma; conductor is an isogeny invariant).", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "-1", "-2", "1", "4", "-1", "-2", "1" ], "h": [ "0" ], "claimed_group": [ 8 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "2560.h.3" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_8.json" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-17 06:29:03 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "0.770000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/8.c.log", "conductor_log": "data/logs/8.c.conductor.log", "conductor_cputime_seconds": 0.22 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "9.a", "group": [ 9 ], "group_key": "9", "group_bracket": "[9]", "order": 9, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "0", "0", "0", "-1" ], "h": [ "1", "1", "0", "1" ], "f_str": "-x^4", "h_str": "x^3 + x + 1", "g": [ "1", "2", "1", "2", "-2", "0", "1" ], "g_str": "x^6 - 2*x^4 + 2*x^3 + x^2 + 2*x + 1", "g_scale": "1", "discriminant": "2920448", "discriminant_digits": 7, "igusa_clebsch": [ "-184", "292", "-51032", "2920448" ], "g2_invariants": [ "-279841/31", "-10051/31", "943/31" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "0", "0", "-1", "-1", "0", "-1" ], "h": [ "1", "0", "1", "1" ] }, "minimal_discriminant": "713", "minimal_discriminant_factored": "23 * 31", "conductor": "713", "conductor_factored": "23 * 31", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "713.b.713.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/713/b/713/1", "snapshot_label": "" }, "torsion": { "invariants": [ 9 ], "order": 9, "generators": [ { "order": 9, "mumford": [ "x^2", "x + 1", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 9, "point_count_primes": [ 3, 5, 7, 11, 13, 17, 19, 29, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 9 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 7, "chi": "T^4 + T^3 - 4*T^2 + 7*T + 49", "primes_tried_below": 500, "admissible_primes": 3, "reducible_chi": 2 }, "q_simple": { "certified": true, "prime": 7, "chi": "T^4 + T^3 - 4*T^2 + 7*T + 49" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 43, "reducible_chi": 12 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 9 (source: \\cite{Flynn1991})", "sources": [ "Flynn1991" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "0", "0", "0", "-1" ], "h": [ "1", "1", "0", "1" ], "claimed_group": [ 9 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "713.b.713.1" }, "source": { "kind": "import", "name": "paper-t1-009", "file": "table1.txt", "line": 15, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:15 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.210000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/9.a.log", "conductor_log": "data/logs/9.a.conductor.log", "conductor_cputime_seconds": 0.19 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "9.b", "group": [ 9 ], "group_key": "9", "group_bracket": "[9]", "order": 9, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-3", "1", "-5", "1", "-2", "-1" ], "h": [ "0", "1", "0", "1" ], "f_str": "-x^5 - 2*x^4 + x^3 - 5*x^2 + x - 3", "h_str": "x^3 + x", "g": [ "-12", "4", "-19", "4", "-6", "-4", "1" ], "g_str": "x^6 - 4*x^5 - 6*x^4 + 4*x^3 - 19*x^2 + 4*x - 12", "g_scale": "1", "discriminant": "1541599428608", "discriminant_digits": 13, "igusa_clebsch": [ "512", "1957120", "559897024", "1541599428608" ], "g2_invariants": [ "134217728/47045881", "-1835008/130321", "-863744/130321" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "-3", "1", "-5", "1", "-2", "-1" ], "h": [ "0", "1", "0", "1" ] }, "minimal_discriminant": "376367048", "minimal_discriminant_factored": "2^3 * 19^6", "conductor": "722", "conductor_factored": "2 * 19^2", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "722.a.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-3%2C1%2C-5%2C1%2C-2%2C-1%5D%2C%5B0%2C1%2C0%2C1%5D%5D", "snapshot_label": "722.a.1" }, "torsion": { "invariants": [ 9 ], "order": 9, "generators": [ { "order": 9, "mumford": [ "x^2 + x + 1", "3*x + 5", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 9, "point_count_primes": [ 3, 5, 7, 11, 13, 17, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 9 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 93, "reducible_chi": 93 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "cover", "over": "Q", "detail": "a nonconstant map of degree 3 from y^2 = g to the genus-1 curve v^2 = (1)u^3 + (0)u^2 + (-12096)u + (-544752) over Q (the elliptic curve 19a1), (x,y) -> (p/q, y*hh/q^2) with p = 192*x^3 - 276*x^2 + 684*x - 84, q = x^3 - 5*x^2 - 4, hh = 2052*x^3 - 4104*x + 4104; degree-3 map to the elliptic curve 19a1 over Q [Genus2Torsion/verify_split_certificates.m, cover 5]" }, "isogenous_product": null, "cover_submitted": true, "cover_accepted": true, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 44, "reducible_chi": 44 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 9 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-3", "1", "-5", "1", "-2", "-1" ], "h": [ "0", "1", "0", "1" ], "claimed_group": [ 9 ], "claimed_class": "split", "cover": { "field": "", "cremona": "19a1", "cubic": [], "p": [ "-84", "684", "-276", "192" ], "q": [ "-4", "0", "-5", "1" ], "hh": [ "4104", "-4104", "0", "2052" ], "note": "degree-3 map to the elliptic curve 19a1 over Q [Genus2Torsion/verify_split_certificates.m, cover 5]" }, "lmfdb_label_given": "722.a.1" }, "source": { "kind": "import", "name": "paper-t2-009", "file": "table2.txt", "line": 15, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:35:56 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.28000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/9.b.log", "conductor_log": "data/logs/9.b.conductor.log", "conductor_cputime_seconds": 0.17 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "9.c", "group": [ 9 ], "group_key": "9", "group_bracket": "[9]", "order": 9, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "26", "3", "-12", "9", "2", "-2", "1" ], "h": [ "0", "1" ], "f_str": "x^6 - 2*x^5 + 2*x^4 + 9*x^3 - 12*x^2 + 3*x + 26", "h_str": "x", "g": [ "104", "12", "-47", "36", "8", "-8", "4" ], "g_str": "4*x^6 - 8*x^5 + 8*x^4 + 36*x^3 - 47*x^2 + 12*x + 104", "g_scale": "1", "discriminant": "-1778384896", "discriminant_digits": 10, "igusa_clebsch": [ "-89888", "506130496", "-11413401371136", "-1778384896" ], "g2_invariants": [ "3299763591802133/8", "-155150527903/4", "-44328573381/2" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "26", "3", "-12", "9", "2", "-2", "1" ], "h": [ "0", "1" ] }, "minimal_discriminant": "-434176", "minimal_discriminant_factored": "-1 * 2^13 * 53", "conductor": "1696", "conductor_factored": "2^5 * 53", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "1696.b.434176.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/1696/b/434176/1", "snapshot_label": "" }, "torsion": { "invariants": [ 9 ], "order": 9, "generators": [ { "order": 9, "mumford": [ "1", "2*x^3 - 2*x^2", "2" ] } ], "method": "generators", "generators_note": "", "generators_submitted": 1, "point_count_gcd": 9, "point_count_primes": [ 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 9 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 93, "reducible_chi": 48 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 - T^2 + 9" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "cover", "over": "X^2 + 1", "detail": "a nonconstant map of degree 3 from y^2 = g to the genus-1 curve v^2 = (-8*w)u^3 + (90*w - 56)u^2 + (36*w + 20)u + (4) over X^2 + 1, (x,y) -> (p/q, y*hh/q^2) with p = x^2 + x - 1/2, q = x^3 + (-3*w - 1)*x^2 + 1/2*(-w - 8)*x + 5*w - 1, hh = x^3 + 1/2*(-3*w + 3)*x^2 + 1/2*(-8*w - 3)*x + 1/4*(-11*w - 9); degree-3 map, defined over Q(sqrt(-1)), from y^2 = g to a genus-1 curve; found by code/gsplit_cover3.m (torsion_jac, 2026-10-16)" }, "isogenous_product": null, "cover_submitted": true, "cover_accepted": true, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 44, "reducible_chi": 23 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": 2016, "historical": false }, "reference": "LMFDB genus 2 curve 1696.b.434176.1, https://www.lmfdb.org/Genus2Curve/Q/1696/b/434176/1", "sources": [ "BSSVY" ], "rm": false, "notes": "First Q-simple, geometrically split example of [9] for the census (the gsplit cell is empty). Found by querying the LMFDB genus 2 data (devmirror.lmfdb.xyz, g2c_curves: is_simple_base = true, is_simple_geom = false); smallest-conductor curve of this torsion group there (LMFDB conductor 1696). The automatic Richelot certificates do not apply (over Q(sqrt(-1)) the working model factors as two cubics and has no Richelot isogeny), so a splitness certificate is supplied: an explicit degree-3 map to a genus-1 curve over Q(sqrt(-1)) (the field over which J splits; the cubic factors of g are the fibre over the point at infinity and the three remaining Weierstrass points). Checked locally with the census's own pipeline/magma/verify_lib.m (Magma V2.30-1). Submitted by F. Najman; the curve is due to the database cited under Reference, the cover was computed for this submission.", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "26", "3", "-12", "9", "2", "-2", "1" ], "h": [ "0", "1" ], "claimed_group": [ 9 ], "claimed_class": "split", "cover": { "field": "X^2+1", "cremona": "", "cubic": [ "-8*w", "90*w-56", "36*w+20", "4" ], "p": [ "-1/2", "1", "1" ], "q": [ "5*w-1", "1/2*(-w-8)", "-3*w-1", "1" ], "hh": [ "1/4*(-11*w-9)", "1/2*(-8*w-3)", "1/2*(-3*w+3)", "1" ], "note": "degree-3 map, defined over Q(sqrt(-1)), from y^2 = g to a genus-1 curve; found by code/gsplit_cover3.m (torsion_jac, 2026-10-16)" }, "lmfdb_label_given": "1696.b.434176.1" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_9.json" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-17 06:29:05 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "0.810000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/9.c.log", "conductor_log": "data/logs/9.c.conductor.log", "conductor_cputime_seconds": 1.27 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "10.a", "group": [ 10 ], "group_key": "10", "group_bracket": "[10]", "order": 10, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "7", "16", "0", "-8", "-2", "1" ], "h": [ "0", "1", "0", "1" ], "f_str": "x^5 - 2*x^4 - 8*x^3 + 16*x + 7", "h_str": "x^3 + x", "g": [ "28", "64", "1", "-32", "-6", "4", "1" ], "g_str": "x^6 + 4*x^5 - 6*x^4 - 32*x^3 + x^2 + 64*x + 28", "g_scale": "1", "discriminant": "1593344", "discriminant_digits": 7, "igusa_clebsch": [ "9760", "817600", "2882646464", "1593344" ], "g2_invariants": [ "2702708163200000/389", "97147868000000/389", "3103255952400/389" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "0", "-1", "-8", "4", "3", "-2" ], "h": [ "0", "1", "0", "1" ] }, "minimal_discriminant": "389", "minimal_discriminant_factored": "389", "conductor": "389", "conductor_factored": "389", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "389.a.389.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/389/a/389/1", "snapshot_label": "" }, "torsion": { "invariants": [ 10 ], "order": 10, "generators": [ { "order": 10, "mumford": [ "x + 2", "x^3 + 8", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 20, "point_count_primes": [ 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101 ], "claimed": [ 10 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 5, "chi": "T^4 - T^3 - 5*T + 25", "primes_tried_below": 500, "admissible_primes": 2, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 + 2*T^3 + 2*T^2 + 6*T + 9" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 45, "reducible_chi": 12 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 10 (source: \\cite{Flynn1991})", "sources": [ "Flynn1991" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "7", "16", "0", "-8", "-2", "1" ], "h": [ "0", "1", "0", "1" ], "claimed_group": [ 10 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "389.a.389.1" }, "source": { "kind": "import", "name": "paper-t1-010", "file": "table1.txt", "line": 16, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:15 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.310000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/10.a.log", "conductor_log": "data/logs/10.a.conductor.log", "conductor_cputime_seconds": 0.1 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "10.b", "group": [ 10 ], "group_key": "10", "group_bracket": "[10]", "order": 10, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-2", "-1", "10", "7", "-13", "-11" ], "h": [ "0", "0", "1" ], "f_str": "-11*x^5 - 13*x^4 + 7*x^3 + 10*x^2 - x - 2", "h_str": "x^2", "g": [ "-8", "-4", "40", "28", "-51", "-44" ], "g_str": "-44*x^5 - 51*x^4 + 28*x^3 + 40*x^2 - 4*x - 8", "g_scale": "1", "discriminant": "-92928", "discriminant_digits": 5, "igusa_clebsch": [ "44384", "409792", "5649542080", "-179908608" ], "g2_invariants": [ "-5256325630316243968/43923", "-1804005053317888/363", "-99735603013264/363" ], "rational_weierstrass_points": 1, "minimal_model": { "f": [ "-2", "11", "-15", "-3", "6", "2" ], "h": [ "0", "0", "1", "1" ] }, "minimal_discriminant": "-43923", "minimal_discriminant_factored": "-1 * 3 * 11^4", "conductor": "363", "conductor_factored": "3 * 11^2", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "363.a.43923.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/363/a/43923/1", "snapshot_label": "" }, "torsion": { "invariants": [ 10 ], "order": 10, "generators": [ { "order": 10, "mumford": [ "x^2 + x - 1", "-5*x + 3", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 40, "point_count_primes": [ 5, 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 10 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "cover", "over": "Q", "detail": "a nonconstant map of degree 3 from y^2 = g to the genus-1 curve v^2 = (1)u^3 + (0)u^2 + (-8451)u + (-296514) over Q (the elliptic curve 33a2), (x,y) -> (p/q, y*hh/q^2) with p = -57*x^3 - 606/11*x^2 + 156/11*x + 192/11, q = x^3 + 10/11*x^2 - 4/11*x - 4/11, hh = 108/11*x^3 + 216/11*x^2 + 1404/121*x + 216/121; degree-3 map to the elliptic curve 33a2 over Q [Genus2Torsion/verify_split_certificates.m, cover 10]" }, "isogenous_product": null, "cover_submitted": true, "cover_accepted": true, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 10 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-2", "-1", "10", "7", "-13", "-11" ], "h": [ "0", "0", "1" ], "claimed_group": [ 10 ], "claimed_class": "split", "cover": { "field": "", "cremona": "33a2", "cubic": [], "p": [ "192/11", "156/11", "-606/11", "-57" ], "q": [ "-4/11", "-4/11", "10/11", "1" ], "hh": [ "216/121", "1404/121", "216/11", "108/11" ], "note": "degree-3 map to the elliptic curve 33a2 over Q [Genus2Torsion/verify_split_certificates.m, cover 10]" }, "lmfdb_label_given": "363.a.43923.1" }, "source": { "kind": "import", "name": "paper-t2-010", "file": "table2.txt", "line": 16, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:35:57 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.11000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/10.b.log", "conductor_log": "data/logs/10.b.conductor.log", "conductor_cputime_seconds": 0.17 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "10.c", "group": [ 10 ], "group_key": "10", "group_bracket": "[10]", "order": 10, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "0", "-1", "0", "-1" ], "h": [ "1", "1", "1", "1" ], "f_str": "-x^3 - x", "h_str": "x^3 + x^2 + x + 1", "g": [ "1", "-2", "3", "0", "3", "2", "1" ], "g_str": "x^6 + 2*x^5 + 3*x^4 + 3*x^2 - 2*x + 1", "g_scale": "1", "discriminant": "-2359296", "discriminant_digits": 7, "igusa_clebsch": [ "-544", "7936", "-1339392", "-2359296" ], "g2_invariants": [ "22717712/9", "540430/9", "-289" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "0", "-1", "0", "-1" ], "h": [ "1", "1", "1", "1" ] }, "minimal_discriminant": "-576", "minimal_discriminant_factored": "-1 * 2^6 * 3^2", "conductor": "576", "conductor_factored": "2^6 * 3^2", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "576.a.576.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/576/a/576/1", "snapshot_label": "" }, "torsion": { "invariants": [ 10 ], "order": 10, "generators": [ { "order": 10, "mumford": [ "x", "-x^3 - 1", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "the submitted generators generate a subgroup of order 10, but the point-count bound is 20; TorsionSubgroup was run", "generators_submitted": 1, "point_count_gcd": 20, "point_count_primes": [ 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 10 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 93, "reducible_chi": 56 }, "q_simple": { "certified": true, "prime": 19, "chi": "T^4 - 22*T^2 + 361" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "richelot", "over": "Q", "detail": "(2,2)-isogenous over Q to a Weil restriction of an elliptic curve E over Q(sqrt(2)) (the surface is E x E^sigma)" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 44, "reducible_chi": 26 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": 2016, "historical": false }, "reference": "LMFDB genus 2 curve 576.a.576.1, https://www.lmfdb.org/Genus2Curve/Q/576/a/576/1", "sources": [ "BSSVY" ], "rm": false, "notes": "First Q-simple, geometrically split example of [10] for the census (the gsplit cell is empty). Found by querying the LMFDB genus 2 data (devmirror.lmfdb.xyz, g2c_curves: is_simple_base = true, is_simple_geom = false, torsion_subgroup = [10]; geom_end_alg = M_2(Q), end_alg = CM (LMFDB notation)); smallest conductor among the curves of this group that the census verifier certifies (576; the extended database also has 576.b.1 with the same conductor). LMFDB conductor 576. Checked locally with the census's own pipeline/magma/verify_lib.m (Magma V2.30-1): J(Q)_tors = [10], class gsplit, Q-simplicity prime 19 (chi = T^4 - 22*T^2 + 361), splitness certificate: richelot over Q ((2,2)-isogenous over Q to a Weil restriction of an elliptic curve E over Q(sqrt(2)) (the surface is E x E^sigma)). Submitted by F. Najman; the curve is due to the database cited under Reference. Rigorous conductor: J is (2,2)-isogenous over Q to Res_(K/Q) E with K the quadratic field of discriminant 8, so N(J) = |d_K|^2 * Norm(f_E) = 64 * 9 = 576 (computed with Magma; conductor is an isogeny invariant).", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "0", "-1", "0", "-1" ], "h": [ "1", "1", "1", "1" ], "claimed_group": [ 10 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "576.a.576.1" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_10.json" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-17 06:01:48 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "0.910000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/10.c.log", "conductor_log": "data/logs/10.c.conductor.log", "conductor_cputime_seconds": 0.16 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "11.a", "group": [ 11 ], "group_key": "11", "group_bracket": "[11]", "order": 11, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "0", "1" ], "h": [ "1", "1", "0", "1" ], "f_str": "x^2", "h_str": "x^3 + x + 1", "g": [ "1", "2", "5", "2", "2", "0", "1" ], "g_str": "x^6 + 2*x^4 + 2*x^3 + 5*x^2 + 2*x + 1", "g_scale": "1", "discriminant": "-1445888", "discriminant_digits": 7, "igusa_clebsch": [ "-376", "3268", "-246968", "-1445888" ], "g2_invariants": [ "229345007/353", "6021734/353", "565504/353" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "0", "0", "0", "-1", "1", "-1" ], "h": [ "1", "0", "1", "1" ] }, "minimal_discriminant": "-353", "minimal_discriminant_factored": "-1 * 353", "conductor": "353", "conductor_factored": "353", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "353.a.353.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/353/a/353/1", "snapshot_label": "" }, "torsion": { "invariants": [ 11 ], "order": 11, "generators": [ { "order": 11, "mumford": [ "x^2", "x + 1", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 11, "point_count_primes": [ 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101 ], "claimed": [ 11 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 3, "chi": "T^4 + 2*T^3 + 4*T^2 + 6*T + 9", "primes_tried_below": 500, "admissible_primes": 1, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 + 2*T^3 + 4*T^2 + 6*T + 9" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 45, "reducible_chi": 8 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Platonov–Petrunin 2012a", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 11 (source: \\cite{Ogg1973})", "sources": [ "Ogg1973" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "0", "1" ], "h": [ "1", "1", "0", "1" ], "claimed_group": [ 11 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "353.a.353.1" }, "source": { "kind": "import", "name": "paper-t1-011", "file": "table1.txt", "line": 17, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:16 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.300000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/11.a.log", "conductor_log": "data/logs/11.a.conductor.log", "conductor_cputime_seconds": 0.13 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "11.b", "group": [ 11 ], "group_key": "11", "group_bracket": "[11]", "order": 11, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "9471400", "94188680", "99564424", "-65405928", "31412066", "-5109634", "1204142" ], "h": [ "0" ], "f_str": "1204142*x^6 - 5109634*x^5 + 31412066*x^4 - 65405928*x^3 + 99564424*x^2 + 94188680*x + 9471400", "h_str": "0", "g": [ "37885600", "376754720", "398257696", "-261623712", "125648264", "-20438536", "4816568" ], "g_str": "4816568*x^6 - 20438536*x^5 + 125648264*x^4 - 261623712*x^3 + 398257696*x^2 + 376754720*x + 37885600", "g_scale": "1", "discriminant": "40362481003722871916398943024250359610553144532744438252830720000000000000000000000", "discriminant_digits": 83, "igusa_clebsch": [ "-741771862733271040", "68835931289544665408400496200515584", "-11833223631748266633450572454673882552189597519970304", "40362481003722871916398943024250359610553144532744438252830720000000000000000000000" ], "g2_invariants": [ "-360191432142940735484084029162709907157/517904210209855242382812500000000", "12026541516488943984249794828813000971571/414323368167884193906250000000000000", "50843593944525540895616548260050688/252882915141530880069732666015625" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "9471400", "94188680", "99564424", "-65405928", "31412066", "-5109634", "1204142" ], "h": [] }, "minimal_discriminant": "9854121338799529276464585699279872951795201301939560120320000000000000000000000", "minimal_discriminant_factored": "2^53 * 5^22 * 11^17 * 23^22", "conductor": "", "conductor_factored": "", "conductor_note": "not computed: Magma's genus-2 Conductor failed on this curve: Fibre blowups over number fields not yet implemented", "conductor_source": "" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B9471400%2C94188680%2C99564424%2C-65405928%2C31412066%2C-5109634%2C1204142%5D%2C%5B0%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 11 ], "order": 11, "generators": [ { "order": 11, "mumford": [ "x^2 + 1654/151*x + 1090/151", "3689246000/22801*x + 2486300000/22801", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 11, "point_count_primes": [ 3, 7, 13, 17, 19, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 11 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 91, "reducible_chi": 50 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 + T^2 + 9" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "richelot", "over": "Q", "detail": "(2,2)-isogenous over Q to a Weil restriction of an elliptic curve E over Q(sqrt(11)) (the surface is E x E^sigma)" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 42, "reducible_chi": 24 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Balakrishnan–Najman–Shnidman–Sutherland 2026", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 11 (source: new)", "sources": [ "BNSS2026" ], "rm": false, "notes": "new: conjugate (2,2)-glue over Q(sqrt 11) of the X1(11) quadratic point s0=4/5; first split realization of [11] (simple-only since Ogg 1973)", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "9471400", "94188680", "99564424", "-65405928", "31412066", "-5109634", "1204142" ], "h": [ "0" ], "claimed_group": [ 11 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "import", "name": "paper-t2-011", "file": "table2.txt", "line": 17, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:35:59 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.18000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/11.b.log", "conductor_log": "data/logs/11.b.conductor.log", "conductor_cputime_seconds": null }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "12.a", "group": [ 12 ], "group_key": "12", "group_bracket": "[12]", "order": 12, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "1", "1", "1" ], "h": [ "0", "1", "1", "1" ], "f_str": "x^2 + x + 1", "h_str": "x^3 + x^2 + x", "g": [ "4", "4", "5", "2", "3", "2", "1" ], "g_str": "x^6 + 2*x^5 + 3*x^4 + 2*x^3 + 5*x^2 + 4*x + 4", "g_scale": "1", "discriminant": "-12484608", "discriminant_digits": 8, "igusa_clebsch": [ "-856", "12676", "-2843896", "-12484608" ], "g2_invariants": [ "14025517307/3048", "140879945/1016", "20871527/3048" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "1", "1", "1" ], "h": [ "0", "1", "1", "1" ] }, "minimal_discriminant": "-3048", "minimal_discriminant_factored": "-1 * 2^3 * 3 * 127", "conductor": "762", "conductor_factored": "2 * 3 * 127", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "762.a.3048.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/762/a/3048/1", "snapshot_label": "" }, "torsion": { "invariants": [ 12 ], "order": 12, "generators": [ { "order": 12, "mumford": [ "x^2", "-x - 2", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 24, "point_count_primes": [ 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 12 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 7, "chi": "T^4 - T^3 + 6*T^2 - 7*T + 49", "primes_tried_below": 500, "admissible_primes": 2, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 5, "chi": "T^4 + 3*T^3 + 4*T^2 + 15*T + 25" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 43, "reducible_chi": 22 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 12 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "1", "1", "1" ], "h": [ "0", "1", "1", "1" ], "claimed_group": [ 12 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "762.a.3048.1" }, "source": { "kind": "import", "name": "paper-t1-012", "file": "table1.txt", "line": 18, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:16 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.260000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/12.a.log", "conductor_log": "data/logs/12.a.conductor.log", "conductor_cputime_seconds": 0.22 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "12.b", "group": [ 12 ], "group_key": "12", "group_bracket": "[12]", "order": 12, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "0", "0", "1", "-1", "1" ], "h": [ "1", "0", "0", "1" ], "f_str": "x^4 - x^3 + x^2", "h_str": "x^3 + 1", "g": [ "1", "0", "4", "-2", "4", "0", "1" ], "g_str": "x^6 + 4*x^4 - 2*x^3 + 4*x^2 + 1", "g_scale": "1", "discriminant": "-1204224", "discriminant_digits": 7, "igusa_clebsch": [ "-472", "2020", "-147608", "-1204224" ], "g2_invariants": [ "714924299/294", "12733498/147", "327214/49" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "0", "0", "1", "-1", "1" ], "h": [ "1", "0", "0", "1" ] }, "minimal_discriminant": "-294", "minimal_discriminant_factored": "-1 * 2 * 3 * 7^2", "conductor": "294", "conductor_factored": "14 * 21 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 14 * 21 for the elliptic curves ['1', '0', '1', '-1', '0'] and ['1', '0', '0', '1', '0'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "294" }, "lmfdb": { "label": "294.a.294.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/294/a/294/1", "snapshot_label": "" }, "torsion": { "invariants": [ 12 ], "order": 12, "generators": [ { "order": 12, "mumford": [ "1", "-x^3", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 48, "point_count_primes": [ 5, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 12 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (z : y : x) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "0", "1", "-1", "0" ], [ "1", "0", "0", "1", "0" ] ], "conductors": [ "14", "21" ], "conductor": "294", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 12 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "0", "1", "-1", "1" ], "h": [ "1", "0", "0", "1" ], "claimed_group": [ 12 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "294.a.294.1" }, "source": { "kind": "import", "name": "paper-t2-012", "file": "table2.txt", "line": 18, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:36:00 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.09000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/12.b.log", "conductor_log": "data/logs/12.b.conductor.log", "conductor_cputime_seconds": 0.25 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "12.c", "group": [ 12 ], "group_key": "12", "group_bracket": "[12]", "order": 12, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "0", "-2", "1", "0", "-1" ], "h": [ "1", "1", "1", "1" ], "f_str": "-x^4 + x^2 - 2*x", "h_str": "x^3 + x^2 + x + 1", "g": [ "1", "-6", "7", "4", "-1", "2", "1" ], "g_str": "x^6 + 2*x^5 - x^4 + 4*x^3 + 7*x^2 - 6*x + 1", "g_scale": "1", "discriminant": "-14680064", "discriminant_digits": 8, "igusa_clebsch": [ "-512", "40192", "-10571776", "-14680064" ], "g2_invariants": [ "2097152/7", "-126976/7", "-11776" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "0", "-2", "1", "0", "-1" ], "h": [ "1", "1", "1", "1" ] }, "minimal_discriminant": "-3584", "minimal_discriminant_factored": "-1 * 2^9 * 7", "conductor": "1792", "conductor_factored": "2^8 * 7", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "1792.a.3584.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/1792/a/3584/1", "snapshot_label": "" }, "torsion": { "invariants": [ 12 ], "order": 12, "generators": [ { "order": 12, "mumford": [ "x^2 + x - 1", "-1", "2" ] } ], "method": "generators", "generators_note": "", "generators_submitted": 1, "point_count_gcd": 12, "point_count_primes": [ 3, 5, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 12 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 93, "reducible_chi": 49 }, "q_simple": { "certified": true, "prime": 5, "chi": "T^4 - 2*T^2 + 25" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "richelot", "over": "Q", "detail": "(2,2)-isogenous over Q to a Weil restriction of an elliptic curve E over Q(sqrt(2)) (the surface is E x E^sigma)" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 44, "reducible_chi": 22 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": 2016, "historical": false }, "reference": "LMFDB genus 2 curve 1792.a.3584.1, https://www.lmfdb.org/Genus2Curve/Q/1792/a/3584/1", "sources": [ "BSSVY" ], "rm": false, "notes": "First Q-simple, geometrically split example of [12] for the census (the gsplit cell is empty). Found by querying the LMFDB genus 2 data (devmirror.lmfdb.xyz, g2c_curves: is_simple_base = true, is_simple_geom = false, torsion_subgroup = [12]; geom_end_alg = Q x Q, end_alg = Q; smallest conductor among the curves of this group that the census verifier certifies). LMFDB conductor 1792. Checked locally with the census's own pipeline/magma/verify_lib.m (Magma V2.30-1): J(Q)_tors = [12], class gsplit, Q-simplicity prime 5 (chi = T^4 - 2*T^2 + 25), splitness certificate: richelot over Q ((2,2)-isogenous over Q to a Weil restriction of an elliptic curve E over Q(sqrt(2)) (the surface is E x E^sigma)). Submitted by F. Najman; the curve is due to the database cited under Reference. Rigorous conductor: J is (2,2)-isogenous over Q to Res_(K/Q) E with K the quadratic field of discriminant 8, so N(J) = |d_K|^2 * Norm(f_E) = 64 * 28 = 1792 (computed with Magma; conductor is an isogeny invariant).", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "0", "-2", "1", "0", "-1" ], "h": [ "1", "1", "1", "1" ], "claimed_group": [ 12 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "1792.a.3584.1" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_12.json" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-17 06:01:49 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "0.770000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/12.c.log", "conductor_log": "data/logs/12.c.conductor.log", "conductor_cputime_seconds": 0.12 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "13.a", "group": [ 13 ], "group_key": "13", "group_bracket": "[13]", "order": 13, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "0", "-1", "-1" ], "h": [ "1", "1", "1", "1" ], "f_str": "-x^3 - x^2", "h_str": "x^3 + x^2 + x + 1", "g": [ "1", "2", "-1", "0", "3", "2", "1" ], "g_str": "x^6 + 2*x^5 + 3*x^4 - x^2 + 2*x + 1", "g_scale": "1", "discriminant": "1429504", "discriminant_digits": 7, "igusa_clebsch": [ "-32", "3328", "-93696", "1429504" ], "g2_invariants": [ "-1024/349", "2176/349", "1984/349" ], "rational_weierstrass_points": 1, "minimal_model": { "f": [ "0", "0", "-1", "-1" ], "h": [ "1", "1", "1", "1" ] }, "minimal_discriminant": "349", "minimal_discriminant_factored": "349", "conductor": "349", "conductor_factored": "349", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "349.a.349.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/349/a/349/1", "snapshot_label": "" }, "torsion": { "invariants": [ 13 ], "order": 13, "generators": [ { "order": 13, "mumford": [ "x^2 + x", "x + 1", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 13, "point_count_primes": [ 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101 ], "claimed": [ 13 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 3, "chi": "T^4 + T^3 - T^2 + 3*T + 9", "primes_tried_below": 500, "admissible_primes": 1, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 + T^3 - T^2 + 3*T + 9" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 45, "reducible_chi": 4 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Flynn 1990; Leprévost 1991 (order 13)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 13 (source: \\cite{Flynn1990,Leprevost1991a})", "sources": [ "Flynn1990", "Leprevost1991a" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "0", "-1", "-1" ], "h": [ "1", "1", "1", "1" ], "claimed_group": [ 13 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "349.a.349.1" }, "source": { "kind": "import", "name": "paper-t1-013", "file": "table1.txt", "line": 19, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:17 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.250000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/13.a.log", "conductor_log": "data/logs/13.a.conductor.log", "conductor_cputime_seconds": 0.1 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "14.a", "group": [ 14 ], "group_key": "14", "group_bracket": "[14]", "order": 14, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "1", "1" ], "h": [ "1", "0", "0", "1" ], "f_str": "x^2 + x", "h_str": "x^3 + 1", "g": [ "1", "4", "4", "2", "0", "0", "1" ], "g_str": "x^6 + 2*x^3 + 4*x^2 + 4*x + 1", "g_scale": "1", "discriminant": "1019904", "discriminant_digits": 7, "igusa_clebsch": [ "-216", "228", "-18072", "1019904" ], "g2_invariants": [ "-4782969/83", "-183708/83", "-7776/83" ], "rational_weierstrass_points": 1, "minimal_model": { "f": [ "0", "0", "0", "-1", "1", "-1" ], "h": [ "1", "0", "0", "1" ] }, "minimal_discriminant": "249", "minimal_discriminant_factored": "3 * 83", "conductor": "249", "conductor_factored": "3 * 83", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "249.a.249.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/249/a/249/1", "snapshot_label": "" }, "torsion": { "invariants": [ 14 ], "order": 14, "generators": [ { "order": 14, "mumford": [ "x", "-x^3 + 1", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 28, "point_count_primes": [ 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 89, 97, 101, 103, 107 ], "claimed": [ 14 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 7, "chi": "T^4 + T^3 - 2*T^2 + 7*T + 49", "primes_tried_below": 500, "admissible_primes": 2, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 5, "chi": "T^4 + 2*T^2 + 25" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 43, "reducible_chi": 12 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 14 (source: \\cite{PP2012,PZP2013})", "sources": [ "PP2012", "PZP2013" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "1", "1" ], "h": [ "1", "0", "0", "1" ], "claimed_group": [ 14 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "249.a.249.1" }, "source": { "kind": "import", "name": "paper-t1-014", "file": "table1.txt", "line": 20, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:17 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.230000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/14.a.log", "conductor_log": "data/logs/14.a.conductor.log", "conductor_cputime_seconds": 0.12 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "14.b", "group": [ 14 ], "group_key": "14", "group_bracket": "[14]", "order": 14, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "0", "-4", "1", "-3", "1", "-1" ], "h": [ "0", "0", "1", "1" ], "f_str": "-x^5 + x^4 - 3*x^3 + x^2 - 4*x", "h_str": "x^3 + x^2", "g": [ "0", "-16", "4", "-12", "5", "-2", "1" ], "g_str": "x^6 - 2*x^5 + 5*x^4 - 12*x^3 + 4*x^2 - 16*x", "g_scale": "1", "discriminant": "4607442944", "discriminant_digits": 10, "igusa_clebsch": [ "1824", "274752", "190167552", "4607442944" ], "g2_invariants": [ "1203384114/2197", "-16111791/2197", "-24663159/4394" ], "rational_weierstrass_points": 1, "minimal_model": { "f": [ "0", "-4", "1", "-3", "1", "-1" ], "h": [ "0", "0", "1", "1" ] }, "minimal_discriminant": "1124864", "minimal_discriminant_factored": "2^9 * 13^3", "conductor": "1352", "conductor_factored": "2^3 * 13^2", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "1352.a.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C-4%2C1%2C-3%2C1%2C-1%5D%2C%5B0%2C0%2C1%2C1%5D%5D", "snapshot_label": "1352.a.1" }, "torsion": { "invariants": [ 14 ], "order": 14, "generators": [ { "order": 14, "mumford": [ "x^2 + 3/2*x + 2", "-17/4*x - 3", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 14, "point_count_primes": [ 3, 5, 7, 11, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 14 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 93, "reducible_chi": 93 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "cover", "over": "Q", "detail": "a nonconstant map of degree 3 from y^2 = g to the genus-1 curve v^2 = (1)u^3 + (0)u^2 + (-3483)u + (121014) over Q (the elliptic curve 26b1), (x,y) -> (p/q, y*hh/q^2) with p = -45*x^3 - 144*x^2 - 36*x - 144, q = x^3 + 4*x, hh = 432*x^3 - 432*x - 864; degree-3 map to the elliptic curve 26b1 over Q [Genus2Torsion/verify_split_certificates.m, cover 3]" }, "isogenous_product": null, "cover_submitted": true, "cover_accepted": true, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 44, "reducible_chi": 44 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 13 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "-4", "1", "-3", "1", "-1" ], "h": [ "0", "0", "1", "1" ], "claimed_group": [ 14 ], "claimed_class": "split", "cover": { "field": "", "cremona": "26b1", "cubic": [], "p": [ "-144", "-36", "-144", "-45" ], "q": [ "0", "4", "0", "1" ], "hh": [ "-864", "-432", "0", "432" ], "note": "degree-3 map to the elliptic curve 26b1 over Q [Genus2Torsion/verify_split_certificates.m, cover 3]" }, "lmfdb_label_given": "1352.a.1" }, "source": { "kind": "import", "name": "paper-t2-013", "file": "table2.txt", "line": 19, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:36:01 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.960000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/14.b.log", "conductor_log": "data/logs/14.b.conductor.log", "conductor_cputime_seconds": 0.15 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "14.c", "group": [ 14 ], "group_key": "14", "group_bracket": "[14]", "order": 14, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "15", "9", "-7", "64", "47", "-27", "3" ], "h": [ "0", "1", "1" ], "f_str": "3*x^6 - 27*x^5 + 47*x^4 + 64*x^3 - 7*x^2 + 9*x + 15", "h_str": "x^2 + x", "g": [ "60", "36", "-27", "258", "189", "-108", "12" ], "g_str": "12*x^6 - 108*x^5 + 189*x^4 + 258*x^3 - 27*x^2 + 36*x + 60", "g_scale": "1", "discriminant": "-69130186764905742336", "discriminant_digits": 20, "igusa_clebsch": [ "152712", "77714178948", "-3619786473943416", "-69130186764905742336" ], "g2_invariants": [ "-10510100501/69984", "8591680039/26244", "-27009585539/118098" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "15", "9", "-7", "64", "47", "-27", "3" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "-16877487003150816", "minimal_discriminant_factored": "-1 * 2^5 * 3^22 * 7^5", "conductor": "9604", "conductor_factored": "2^2 * 7^4", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "9604.b.3", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B15%2C9%2C-7%2C64%2C47%2C-27%2C3%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "9604.b.3" }, "torsion": { "invariants": [ 14 ], "order": 14, "generators": [ { "order": 14, "mumford": [ "x^2 - 7*x + 5", "36*x - 45", "2" ] } ], "method": "generators", "generators_note": "", "generators_submitted": 1, "point_count_gcd": 14, "point_count_primes": [ 5, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 14 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 49 }, "q_simple": { "certified": true, "prime": 5, "chi": "T^4 + 2*T^2 + 25" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "richelot", "over": "Q(sqrt(-7))", "detail": "a chain of at most 2 (2,2)-isogenies over Q(sqrt(-7)) reaches a product of two elliptic curves over Q(sqrt(-7))" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 23 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": 2026, "historical": false }, "reference": "LMFDB extended genus 2 database (Booker–Sutherland), curve 9604.b.3 (table g2c_curves_new, label 9604.b3 on devmirror.lmfdb.xyz, read 2026-10-16)", "sources": [ "BookerSutherland" ], "rm": false, "notes": "First Q-simple, geometrically split example of [14] for the census (the gsplit cell is empty). Found by querying the LMFDB genus 2 data (devmirror.lmfdb.xyz, g2c_curves_new: is_simple_base = true, is_simple_geom = false, torsion_subgroup = [14]; geom_end_alg = M_2(Q), end_alg = RM; smallest conductor among the curves of this group that the census verifier certifies). LMFDB conductor 9604. Checked locally with the census's own pipeline/magma/verify_lib.m (Magma V2.30-1): J(Q)_tors = [14], class gsplit, Q-simplicity prime 5 (chi = T^4 + 2*T^2 + 25), splitness certificate: richelot over Q(sqrt(-7)) (a chain of at most 2 (2,2)-isogenies over Q(sqrt(-7)) reaches a product of two elliptic curves over Q(sqrt(-7))). Submitted by F. Najman; the curve is due to the database cited under Reference.", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "15", "9", "-7", "64", "47", "-27", "3" ], "h": [ "0", "1", "1" ], "claimed_group": [ 14 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "9604.b.3" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_14.json" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-17 06:01:50 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "0.900000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/14.c.log", "conductor_log": "data/logs/14.c.conductor.log", "conductor_cputime_seconds": 0.34 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "15.a", "group": [ 15 ], "group_key": "15", "group_bracket": "[15]", "order": 15, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "-1", "-1" ], "h": [ "1", "1", "1", "1" ], "f_str": "-x^2 - x", "h_str": "x^3 + x^2 + x + 1", "g": [ "1", "-2", "-1", "4", "3", "2", "1" ], "g_str": "x^6 + 2*x^5 + 3*x^4 + 4*x^3 - x^2 - 2*x + 1", "g_scale": "1", "discriminant": "1134592", "discriminant_digits": 7, "igusa_clebsch": [ "-256", "5632", "-611328", "1134592" ], "g2_invariants": [ "-33554432/277", "524288/277", "475136/277" ], "rational_weierstrass_points": 1, "minimal_model": { "f": [ "0", "-1", "-1" ], "h": [ "1", "1", "1", "1" ] }, "minimal_discriminant": "277", "minimal_discriminant_factored": "277", "conductor": "277", "conductor_factored": "277", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "277.a.277.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/277/a/277/1", "snapshot_label": "" }, "torsion": { "invariants": [ 15 ], "order": 15, "generators": [ { "order": 15, "mumford": [ "1", "x^3 + x^2", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 15, "point_count_primes": [ 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101 ], "claimed": [ 15 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 3, "chi": "T^4 + T^3 + T^2 + 3*T + 9", "primes_tried_below": 500, "admissible_primes": 1, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 + T^3 + T^2 + 3*T + 9" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 45, "reducible_chi": 14 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 15 (source: \\cite{Leprevost1991})", "sources": [ "Leprevost1991" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "-1", "-1" ], "h": [ "1", "1", "1", "1" ], "claimed_group": [ 15 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "277.a.277.1" }, "source": { "kind": "import", "name": "paper-t1-015", "file": "table1.txt", "line": 21, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:18 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.260000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/15.a.log", "conductor_log": "data/logs/15.a.conductor.log", "conductor_cputime_seconds": 0.1 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "15.b", "group": [ 15 ], "group_key": "15", "group_bracket": "[15]", "order": 15, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "1", "0", "2", "0", "1" ], "h": [ "0", "0", "0", "1" ], "f_str": "x^4 + 2*x^2 + 1", "h_str": "x^3", "g": [ "4", "0", "8", "0", "4", "0", "1" ], "g_str": "x^6 + 4*x^4 + 8*x^2 + 4", "g_scale": "1", "discriminant": "-7929856", "discriminant_digits": 7, "igusa_clebsch": [ "-1472", "2368", "-369152", "-7929856" ], "g2_invariants": [ "13181630464/121", "49057344/11", "31824640/121" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "1", "0", "2", "0", "1" ], "h": [ "0", "0", "0", "1" ] }, "minimal_discriminant": "-1936", "minimal_discriminant_factored": "-1 * 2^4 * 11^2", "conductor": "484", "conductor_factored": "11 * 44 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 11 * 44 for the elliptic curves ['0', '-1', '1', '0', '0'] and ['0', '1', '0', '3', '-1'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "484" }, "lmfdb": { "label": "484.a.1936.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/484/a/1936/1", "snapshot_label": "" }, "torsion": { "invariants": [ 15 ], "order": 15, "generators": [ { "order": 15, "mumford": [ "x", "x^3 - 2", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 15, "point_count_primes": [ 3, 5, 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 15 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 93, "reducible_chi": 93 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x : y : -z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "0", "-1", "1", "0", "0" ], [ "0", "1", "0", "3", "-1" ] ], "conductors": [ "11", "44" ], "conductor": "484", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 44, "reducible_chi": 44 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 14 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "1", "0", "2", "0", "1" ], "h": [ "0", "0", "0", "1" ], "claimed_group": [ 15 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "484.a.1936.1" }, "source": { "kind": "import", "name": "paper-t2-014", "file": "table2.txt", "line": 20, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:36:03 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.17000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/15.b.log", "conductor_log": "data/logs/15.b.conductor.log", "conductor_cputime_seconds": 0.19 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "15.c", "group": [ 15 ], "group_key": "15", "group_bracket": "[15]", "order": 15, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "1", "-18", "72", "47", "38", "9", "1" ], "h": [ "0", "0", "1" ], "f_str": "x^6 + 9*x^5 + 38*x^4 + 47*x^3 + 72*x^2 - 18*x + 1", "h_str": "x^2", "g": [ "4", "-72", "288", "188", "153", "36", "4" ], "g_str": "4*x^6 + 36*x^5 + 153*x^4 + 188*x^3 + 288*x^2 - 72*x + 4", "g_scale": "1", "discriminant": "21693086377017606144", "discriminant_digits": 20, "igusa_clebsch": [ "-600480", "9125960256", "-1735368357210624", "21693086377017606144" ], "g2_invariants": [ "-162152639684375/360448", "-502546455375/45056", "310585075/5632" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "1", "-18", "72", "47", "38", "9", "1" ], "h": [ "0", "0", "1" ] }, "minimal_discriminant": "5296163666264064", "minimal_discriminant_factored": "2^25 * 3^15 * 11", "conductor": "3168", "conductor_factored": "2^5 * 3^2 * 11", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "3168.h.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B1%2C-18%2C72%2C47%2C38%2C9%2C1%5D%2C%5B0%2C0%2C1%5D%5D", "snapshot_label": "3168.h.1" }, "torsion": { "invariants": [ 15 ], "order": 15, "generators": [ { "order": 15, "mumford": [ "1", "-2*x^3 - 9*x^2", "2" ] } ], "method": "generators", "generators_note": "", "generators_submitted": 1, "point_count_gcd": 15, "point_count_primes": [ 5, 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 15 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 52 }, "q_simple": { "certified": true, "prime": 5, "chi": "T^4 + 4*T^2 + 25" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "cover", "over": "X^2 - 3", "detail": "a nonconstant map of degree 3 from y^2 = g to the genus-1 curve v^2 = (-432*w + 864)u^3 + (36*w + 45)u^2 + (24*w - 12)u + (4) over X^2 - 3, (x,y) -> (p/q, y*hh/q^2) with p = x^2 + (-w + 1)*x, q = x^3 + 1/2*(-4*w + 9)*x^2 + (-7*w + 15)*x + w - 2, hh = x^3 + (-w + 3)*x^2 + 1/2*(5*w - 9)*x - 4*w + 7; degree-3 map, defined over Q(sqrt(3)), from y^2 = g to a genus-1 curve; found by code/gsplit_cover3.m (torsion_jac, 2026-10-16)" }, "isogenous_product": null, "cover_submitted": true, "cover_accepted": true, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 25 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": 2026, "historical": false }, "reference": "LMFDB extended genus 2 database (Booker–Sutherland), curve 3168.h.1 (table g2c_curves_new, label 3168.h1 on devmirror.lmfdb.xyz, read 2026-10-16)", "sources": [ "BookerSutherland" ], "rm": false, "notes": "First Q-simple, geometrically split example of [15] for the census (the gsplit cell is empty). Found by querying the LMFDB genus 2 data (devmirror.lmfdb.xyz, g2c_curves_new: is_simple_base = true, is_simple_geom = false); smallest-conductor curve of this torsion group there (LMFDB conductor 3168). The automatic Richelot certificates do not apply (over Q(sqrt(3)) the working model factors as two cubics and has no Richelot isogeny), so a splitness certificate is supplied: an explicit degree-3 map to a genus-1 curve over Q(sqrt(3)) (the field over which J splits; the cubic factors of g are the fibre over the point at infinity and the three remaining Weierstrass points). Checked locally with the census's own pipeline/magma/verify_lib.m (Magma V2.30-1). Submitted by F. Najman; the curve is due to the database cited under Reference, the cover was computed for this submission.", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "1", "-18", "72", "47", "38", "9", "1" ], "h": [ "0", "0", "1" ], "claimed_group": [ 15 ], "claimed_class": "split", "cover": { "field": "X^2-3", "cremona": "", "cubic": [ "-432*w+864", "36*w+45", "24*w-12", "4" ], "p": [ "0", "-w+1", "1" ], "q": [ "w-2", "-7*w+15", "1/2*(-4*w+9)", "1" ], "hh": [ "-4*w+7", "1/2*(5*w-9)", "-w+3", "1" ], "note": "degree-3 map, defined over Q(sqrt(3)), from y^2 = g to a genus-1 curve; found by code/gsplit_cover3.m (torsion_jac, 2026-10-16)" }, "lmfdb_label_given": "3168.h.1" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_15.json" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-17 06:01:53 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "0.800000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/15.c.log", "conductor_log": "data/logs/15.c.conductor.log", "conductor_cputime_seconds": 2.19 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "16.a", "group": [ 16 ], "group_key": "16", "group_bracket": "[16]", "order": 16, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "1", "1", "-2", "0", "1", "-1" ], "h": [ "1", "0", "0", "1" ], "f_str": "-x^5 + x^4 - 2*x^2 + x + 1", "h_str": "x^3 + 1", "g": [ "5", "4", "-8", "2", "4", "-4", "1" ], "g_str": "x^6 - 4*x^5 + 4*x^4 + 2*x^3 - 8*x^2 + 4*x + 5", "g_scale": "1", "discriminant": "-27197440", "discriminant_digits": 8, "igusa_clebsch": [ "-1304", "17092", "-10717752", "-27197440" ], "g2_invariants": [ "115063617043/6640", "4023263963/6640", "-13842449/6640" ], "rational_weierstrass_points": 1, "minimal_model": { "f": [ "1", "-1", "-1", "-1", "-1" ], "h": [ "0", "1", "1", "1" ] }, "minimal_discriminant": "-6640", "minimal_discriminant_factored": "-1 * 2^4 * 5 * 83", "conductor": "830", "conductor_factored": "2 * 5 * 83", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "830.a.6640.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/830/a/6640/1", "snapshot_label": "" }, "torsion": { "invariants": [ 16 ], "order": 16, "generators": [ { "order": 16, "mumford": [ "x^2 - 1", "x + 1", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 16, "point_count_primes": [ 3, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 89, 97, 101, 103, 107 ], "claimed": [ 16 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 3, "chi": "T^4 + T^3 + 2*T^2 + 3*T + 9", "primes_tried_below": 500, "admissible_primes": 1, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 + T^3 + 2*T^2 + 3*T + 9" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 43, "reducible_chi": 12 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 16 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "1", "1", "-2", "0", "1", "-1" ], "h": [ "1", "0", "0", "1" ], "claimed_group": [ 16 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "830.a.6640.1" }, "source": { "kind": "import", "name": "paper-t1-016", "file": "table1.txt", "line": 22, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:18 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.300000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/16.a.log", "conductor_log": "data/logs/16.a.conductor.log", "conductor_cputime_seconds": 0.23 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "16.b", "group": [ 16 ], "group_key": "16", "group_bracket": "[16]", "order": 16, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-125", "0", "97", "0", "-26", "0", "2" ], "h": [ "0", "1", "0", "1" ], "f_str": "2*x^6 - 26*x^4 + 97*x^2 - 125", "h_str": "x^3 + x", "g": [ "-500", "0", "389", "0", "-102", "0", "9" ], "g_str": "9*x^6 - 102*x^4 + 389*x^2 - 500", "g_scale": "1", "discriminant": "3732480000000", "discriminant_digits": 13, "igusa_clebsch": [ "1714848", "10854144", "6197659792128", "3732480000000" ], "g2_invariants": [ "116399727907379248994752/234375", "193988089667923989064/9375", "28735596382851136/25" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "-125", "0", "97", "0", "-26", "0", "2" ], "h": [ "0", "1", "0", "1" ] }, "minimal_discriminant": "911250000", "minimal_discriminant_factored": "2^4 * 3^6 * 5^7", "conductor": "2700", "conductor_factored": "15 * 180 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 15 * 180 for the elliptic curves ['1', '1', '1', '35', '-28'] and ['0', '0', '0', '33', '-74'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "2700" }, "lmfdb": { "label": "2700.d.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-125%2C0%2C97%2C0%2C-26%2C0%2C2%5D%2C%5B0%2C1%2C0%2C1%5D%5D", "snapshot_label": "2700.d.1" }, "torsion": { "invariants": [ 16 ], "order": 16, "generators": [ { "order": 16, "mumford": [ "x^2 + 1/3*x - 5", "5/3*x - 5", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 48, "point_count_primes": [ 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 16 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x : y : -z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "1", "1", "35", "-28" ], [ "0", "0", "0", "33", "-74" ] ], "conductors": [ "15", "180" ], "conductor": "2700", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 15 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-125", "0", "97", "0", "-26", "0", "2" ], "h": [ "0", "1", "0", "1" ], "claimed_group": [ 16 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "2700.d.1" }, "source": { "kind": "import", "name": "paper-t2-015", "file": "table2.txt", "line": 21, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:36:04 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.07000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/16.b.log", "conductor_log": "data/logs/16.b.conductor.log", "conductor_cputime_seconds": 0.21 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "17.a", "group": [ 17 ], "group_key": "17", "group_bracket": "[17]", "order": 17, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "1", "3", "1", "1" ], "h": [ "0", "1", "1", "1" ], "f_str": "x^3 + x^2 + 3*x + 1", "h_str": "x^3 + x^2 + x", "g": [ "4", "12", "5", "6", "3", "2", "1" ], "g_str": "x^6 + 2*x^5 + 3*x^4 + 6*x^3 + 5*x^2 + 12*x + 4", "g_scale": "1", "discriminant": "2092957696", "discriminant_digits": 10, "igusa_clebsch": [ "-24", "40548", "8070888", "2092957696" ], "g2_invariants": [ "-243/510976", "5697/255488", "-32319/127744" ], "rational_weierstrass_points": 1, "minimal_model": { "f": [ "1", "3", "1", "1" ], "h": [ "0", "1", "1", "1" ] }, "minimal_discriminant": "510976", "minimal_discriminant_factored": "2^10 * 499", "conductor": "1996", "conductor_factored": "2^2 * 499", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "1996.b.510976.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/1996/b/510976/1", "snapshot_label": "" }, "torsion": { "invariants": [ 17 ], "order": 17, "generators": [ { "order": 17, "mumford": [ "x", "x^3 + 2", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 17, "point_count_primes": [ 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101 ], "claimed": [ 17 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 3, "chi": "T^4 + T^3 + 3*T^2 + 3*T + 9", "primes_tried_below": 500, "admissible_primes": 1, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 + T^3 + 3*T^2 + 3*T + 9" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 45, "reducible_chi": 6 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Leprévost 1991 (orders 15, 17, 19, 21)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 17 (source: \\cite{Leprevost1991})", "sources": [ "Leprevost1991" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "1", "3", "1", "1" ], "h": [ "0", "1", "1", "1" ], "claimed_group": [ 17 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "1996.b.510976.1" }, "source": { "kind": "import", "name": "paper-t1-017", "file": "table1.txt", "line": 23, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:19 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.320000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/17.a.log", "conductor_log": "data/logs/17.a.conductor.log", "conductor_cputime_seconds": 0.17 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "18.a", "group": [ 18 ], "group_key": "18", "group_bracket": "[18]", "order": 18, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "2", "4", "0", "-2" ], "h": [ "1", "0", "0", "1" ], "f_str": "-2*x^4 + 4*x^2 + 2*x", "h_str": "x^3 + 1", "g": [ "1", "8", "16", "2", "-8", "0", "1" ], "g_str": "x^6 - 8*x^4 + 2*x^3 + 16*x^2 + 8*x + 1", "g_scale": "1", "discriminant": "-77332480", "discriminant_digits": 8, "igusa_clebsch": [ "1832", "93028", "47683816", "-77332480" ], "g2_invariants": [ "-629763392149/18880", "-228170791/295", "-5453864/295" ], "rational_weierstrass_points": 1, "minimal_model": { "f": [ "0", "0", "-2", "-1", "4", "-2" ], "h": [ "1", "0", "0", "1" ] }, "minimal_discriminant": "-18880", "minimal_discriminant_factored": "-1 * 2^6 * 5 * 59", "conductor": "1180", "conductor_factored": "2^2 * 5 * 59", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "1180.a.18880.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/1180/a/18880/1", "snapshot_label": "" }, "torsion": { "invariants": [ 18 ], "order": 18, "generators": [ { "order": 18, "mumford": [ "x^2 - 2*x - 1", "3*x + 1", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 18, "point_count_primes": [ 3, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 18 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 7, "chi": "T^4 + T^3 - 4*T^2 + 7*T + 49", "primes_tried_below": 500, "admissible_primes": 2, "reducible_chi": 1 }, "q_simple": { "certified": true, "prime": 7, "chi": "T^4 + T^3 - 4*T^2 + 7*T + 49" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 43, "reducible_chi": 12 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 18 (source: \\cite{PP2012,PZP2013})", "sources": [ "PP2012", "PZP2013" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "2", "4", "0", "-2" ], "h": [ "1", "0", "0", "1" ], "claimed_group": [ 18 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "1180.a.18880.1" }, "source": { "kind": "import", "name": "paper-t1-018", "file": "table1.txt", "line": 24, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:19 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.250000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/18.a.log", "conductor_log": "data/logs/18.a.conductor.log", "conductor_cputime_seconds": 0.24 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "18.b", "group": [ 18 ], "group_key": "18", "group_bracket": "[18]", "order": 18, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-6", "5", "-7", "2", "-1", "-1" ], "h": [ "0", "0", "1", "1" ], "f_str": "-x^5 - x^4 + 2*x^3 - 7*x^2 + 5*x - 6", "h_str": "x^3 + x^2", "g": [ "-24", "20", "-28", "8", "-3", "-2", "1" ], "g_str": "x^6 - 2*x^5 - 3*x^4 + 8*x^3 - 28*x^2 + 20*x - 24", "g_scale": "1", "discriminant": "9637806080", "discriminant_digits": 10, "igusa_clebsch": [ "3200", "1298560", "1852825024", "9637806080" ], "g2_invariants": [ "512000000000/117649", "-64000000/343", "-255608000/2401" ], "rational_weierstrass_points": 1, "minimal_model": { "f": [ "-6", "5", "-7", "2", "-1", "-1" ], "h": [ "0", "0", "1", "1" ] }, "minimal_discriminant": "2352980", "minimal_discriminant_factored": "2^2 * 5 * 7^6", "conductor": "490", "conductor_factored": "2 * 5 * 7^2", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "490.a.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-6%2C5%2C-7%2C2%2C-1%2C-1%5D%2C%5B0%2C0%2C1%2C1%5D%5D", "snapshot_label": "490.a.1" }, "torsion": { "invariants": [ 18 ], "order": 18, "generators": [ { "order": 18, "mumford": [ "x^2 - 1/2*x + 3/2", "-1/4*x - 3/4", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 18, "point_count_primes": [ 3, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 18 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "cover", "over": "Q", "detail": "a nonconstant map of degree 3 from y^2 = g to the genus-1 curve v^2 = (1)u^3 + (0)u^2 + (11232)u + (-78192) over Q (the elliptic curve 35a1), (x,y) -> (p/q, y*hh/q^2) with p = 48*x^3 + 60*x^2 - 12*x - 36, q = x^3 - 4*x^2 + 5*x - 6, hh = 756*x^3 - 2268*x^2 + 3780*x - 756; degree-3 map to the elliptic curve 35a1 over Q [Genus2Torsion/verify_split_certificates.m, cover 6]" }, "isogenous_product": null, "cover_submitted": true, "cover_accepted": true, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 16 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-6", "5", "-7", "2", "-1", "-1" ], "h": [ "0", "0", "1", "1" ], "claimed_group": [ 18 ], "claimed_class": "split", "cover": { "field": "", "cremona": "35a1", "cubic": [], "p": [ "-36", "-12", "60", "48" ], "q": [ "-6", "5", "-4", "1" ], "hh": [ "-756", "3780", "-2268", "756" ], "note": "degree-3 map to the elliptic curve 35a1 over Q [Genus2Torsion/verify_split_certificates.m, cover 6]" }, "lmfdb_label_given": "490.a.1" }, "source": { "kind": "import", "name": "paper-t2-016", "file": "table2.txt", "line": 22, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:36:06 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.08000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/18.b.log", "conductor_log": "data/logs/18.b.conductor.log", "conductor_cputime_seconds": 0.22 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "18.c", "group": [ 18 ], "group_key": "18", "group_bracket": "[18]", "order": 18, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "10", "-16", "-14", "5", "0", "-1" ], "h": [ "0", "1", "0", "1" ], "f_str": "-x^5 + 5*x^3 - 14*x^2 - 16*x + 10", "h_str": "x^3 + x", "g": [ "40", "-64", "-55", "20", "2", "-4", "1" ], "g_str": "x^6 - 4*x^5 + 2*x^4 + 20*x^3 - 55*x^2 - 64*x + 40", "g_scale": "1", "discriminant": "-281792804290560", "discriminant_digits": 15, "igusa_clebsch": [ "4800", "-21784320", "-70418681856", "-281792804290560" ], "g2_invariants": [ "-1953125/1728", "-109375/144", "-4000/9" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "10", "-16", "-14", "5", "0", "-1" ], "h": [ "0", "1", "0", "1" ] }, "minimal_discriminant": "-68797071360", "minimal_discriminant_factored": "-1 * 2^21 * 3^8 * 5", "conductor": "17280", "conductor_factored": "2^7 * 3^3 * 5", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "17280.z.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B10%2C-16%2C-14%2C5%2C0%2C-1%5D%2C%5B0%2C1%2C0%2C1%5D%5D", "snapshot_label": "17280.z.1" }, "torsion": { "invariants": [ 18 ], "order": 18, "generators": [ { "order": 18, "mumford": [ "x^2 - 4*x + 1", "-6*x + 6", "2" ] } ], "method": "generators", "generators_note": "", "generators_submitted": 1, "point_count_gcd": 18, "point_count_primes": [ 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 18 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 51 }, "q_simple": { "certified": true, "prime": 7, "chi": "T^4 + 4*T^2 + 49" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "cover", "over": "X^2 - 6", "detail": "a nonconstant map of degree 3 from y^2 = g to the genus-1 curve v^2 = (-20*w - 48)u^3 + (-18*w - 41)u^2 + (-2*w - 2)u + (1) over X^2 - 6, (x,y) -> (p/q, y*hh/q^2) with p = x^2 + (-2*w + 2)*x - 2*w - 5, q = x^3 - 2*x^2 + (2*w - 1)*x + 2*w + 8, hh = x^3 + (-w - 3)*x^2 + (-6*w - 15)*x + w + 1; degree-3 map, defined over Q(sqrt(6)), from y^2 = g to a genus-1 curve; found by code/gsplit2_cover3.m (torsion_jac, 2026-10-17)" }, "isogenous_product": null, "cover_submitted": true, "cover_accepted": true, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 25 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": 2026, "historical": false }, "reference": "LMFDB extended genus 2 database (Booker–Sutherland), curve 17280.z.1 (table g2c_curves_new, label 17280.z1 on devmirror.lmfdb.xyz, read 2026-10-17)", "sources": [ "BookerSutherland" ], "rm": false, "notes": "First Q-simple, geometrically split example of [18] for the census (the gsplit cell is empty). Found by querying the LMFDB genus 2 data (devmirror.lmfdb.xyz, both g2c_curves and g2c_curves_new, all curves with is_simple_base = true, is_simple_geom = false): there are exactly three such curves with torsion [18], 17280.z1, 36000.e1 and 36000.e2, all in the extended table; this is the one of smallest conductor. J splits over Q(sqrt(6)) as E x E^sigma (E in the isogeny class 2.2.24.1-30.1-b, not a Q-curve). The automatic Richelot certificates do not apply (over Q(sqrt(6)) the working model factors as 1+1+2+2 and its only Richelot neighbour leads to no split surface), so a splitness certificate is supplied: an explicit degree-3 map to a genus-1 curve over Q(sqrt(6)); the fibre over the point at infinity is the cubic (linear)*(quadratic) factor of g over Q(sqrt(6)). Checked locally with the census's own pipeline/magma/verify_lib.m (Magma V2.30-1): J(Q)_tors = [18] from the submitted generator (gcd #J(F_p) = 18), Q-simple (chi_7 irreducible), class gsplit. Conductor 17280 = 2^7 * 3^3 * 5: Magma's genus-2 Conductor uses Ogg's formula at 2 here (v_2(disc) >= 12), but the value is rigorous: J is Q-isogenous to Res_{K/Q} E_T, E_T the target of the cover, so N(J) = |d_K|^2 Norm(f_{E_T}) = 576 * 30 = 17280, which agrees with the LMFDB. Submitted by F. Najman; the curve is due to the database cited under Reference, the cover was computed for this submission.", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "10", "-16", "-14", "5", "0", "-1" ], "h": [ "0", "1", "0", "1" ], "claimed_group": [ 18 ], "claimed_class": "split", "cover": { "field": "X^2-6", "cremona": "", "cubic": [ "-20*w-48", "-18*w-41", "-2*w-2", "1" ], "p": [ "-2*w-5", "-2*w+2", "1" ], "q": [ "2*w+8", "2*w-1", "-2", "1" ], "hh": [ "w+1", "-6*w-15", "-w-3", "1" ], "note": "degree-3 map, defined over Q(sqrt(6)), from y^2 = g to a genus-1 curve; found by code/gsplit2_cover3.m (torsion_jac, 2026-10-17)" }, "lmfdb_label_given": "17280.z.1" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_18.json" }, "dates": { "submitted": "2026-10-17", "verified": "2026-10-17 10:20:31 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "1.57000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/18.c.log", "conductor_log": "data/logs/18.c.conductor.log", "conductor_cputime_seconds": 2.67 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "19.a", "group": [ 19 ], "group_key": "19", "group_bracket": "[19]", "order": 19, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "-2", "5", "-3", "-5", "1", "1", "1" ], "h": [ "1", "1", "1" ], "f_str": "x^6 + x^5 + x^4 - 5*x^3 - 3*x^2 + 5*x - 2", "h_str": "x^2 + x + 1", "g": [ "-7", "22", "-9", "-18", "5", "4", "4" ], "g_str": "4*x^6 + 4*x^5 + 5*x^4 - 18*x^3 - 9*x^2 + 22*x - 7", "g_scale": "1", "discriminant": "24628953088", "discriminant_digits": 11, "igusa_clebsch": [ "12904", "-381020", "-1896522840", "24628953088" ], "g2_invariants": [ "10918722712059293/6012928", "58950390267659/751616", "456477769281/93952" ], "rational_weierstrass_points": 1, "minimal_model": { "f": [ "-2", "-5", "-1", "6", "3", "-2" ], "h": [ "1", "0", "0", "1" ] }, "minimal_discriminant": "6012928", "minimal_discriminant_factored": "2^14 * 367", "conductor": "1468", "conductor_factored": "2^2 * 367", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "1468.b.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-2%2C5%2C-3%2C-5%2C1%2C1%2C1%5D%2C%5B1%2C1%2C1%5D%5D", "snapshot_label": "1468.b.1" }, "torsion": { "invariants": [ 19 ], "order": 19, "generators": [ { "order": 19, "mumford": [ "x - 1", "2*x^3 - 3", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 19, "point_count_primes": [ 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101 ], "claimed": [ 19 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 3, "chi": "T^4 + T^3 + 5*T^2 + 3*T + 9", "primes_tried_below": 500, "admissible_primes": 1, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 + T^3 + 5*T^2 + 3*T + 9" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 45, "reducible_chi": 7 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 19 (source: \\cite{Leprevost1991,PZP2013})", "sources": [ "Leprevost1991", "PZP2013" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-2", "5", "-3", "-5", "1", "1", "1" ], "h": [ "1", "1", "1" ], "claimed_group": [ 19 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "1468.b.1" }, "source": { "kind": "import", "name": "paper-t1-019", "file": "table1.txt", "line": 25, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:20 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.330000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/19.a.log", "conductor_log": "data/logs/19.a.conductor.log", "conductor_cputime_seconds": 0.24 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "19.b", "group": [ 19 ], "group_key": "19", "group_bracket": "[19]", "order": 19, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "0", "1", "1" ], "h": [ "1", "0", "1", "1" ], "f_str": "x^2 + x", "h_str": "x^3 + x^2 + 1", "g": [ "1", "4", "6", "2", "1", "2", "1" ], "g_str": "x^6 + 2*x^5 + x^4 + 2*x^3 + 6*x^2 + 4*x + 1", "g_scale": "1", "discriminant": "-692224", "discriminant_digits": 6, "igusa_clebsch": [ "8", "3172", "30056", "-692224" ], "g2_invariants": [ "-1/169", "33/169", "43/169" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "0", "1", "1" ], "h": [ "1", "0", "1", "1" ] }, "minimal_discriminant": "-169", "minimal_discriminant_factored": "-1 * 13^2", "conductor": "169", "conductor_factored": "13^2", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "169.a.169.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/169/a/169/1", "snapshot_label": "" }, "torsion": { "invariants": [ 19 ], "order": 19, "generators": [ { "order": 19, "mumford": [ "x^2 + x", "1", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 19, "point_count_primes": [ 3, 5, 7, 11, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 19 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 93, "reducible_chi": 23 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 + 2*T^3 + T^2 + 6*T + 9" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "cover", "over": "X^6 + 2925/2*X^5 - 13931515/144*X^4 + 555105005/864*X^3 + 741300755/1152*X^2 - 59482995065/62208*X - 2129491223327/2985984", "detail": "a nonconstant map of degree 2 from y^2 = g to the genus-1 curve v^2 = (4)u^3 + (0)u^2 + (-w)u + (1/4205989997374997986989046056*(272112803100608279040*w^5 + 395664740768467375261056*w^4 - 29647589653215861439432320*w^3 + 477245639866772050871629032*w^2 + 195733771623786479752709040*w + 207843556759852706226835301)) over X^6 + 2925/2*X^5 - 13931515/144*X^4 + 555105005/864*X^3 + 741300755/1152*X^2 - 59482995065/62208*X - 2129491223327/2985984, (x,y) -> (p/q, y*hh/q^2) with p = 1/1401996665791665995663015352*(15990842261263127814144*w^5 + 23367448923609665432475648*w^4 - 1575054550234548111640212288*w^3 + 12162386878499496157273057680*w^2 - 4320129434415791253147032064*w - 8711193399793278371297110063)*x^2 + 1/175249583223958249457876919*(1181047455487212756480*w^5 + 1725717953028064158570240*w^4 - 116547640063212079646312544*w^3 + 913046320206876174333647256*w^2 - 421538903357027186361600372*w - 655021524942587374393280381)*x + 1/350499166447916498915753838*(-1047400442066319243264*w^5 - 1530244557357870304783296*w^4 + 103639941866199892102086384*w^3 - 827402670280905916142666448*w^2 + 497327212158924622832114979*w + 825849916525810185225936473), q = x^2 + 1/1518829721274304828634933298*(16541103711402026827776*w^5 + 24166552645798839366764544*w^4 - 1636558488650062614742432704*w^3 + 13063575221221494738458433744*w^2 - 7641246715946420125902252744*w - 11882728243714815255723273527)*x + 1/1518829721274304828634933298*(62752690556056115767296*w^5 + 91701242410418949541259520*w^4 - 6180068925644118438615566976*w^3 + 47674259218659743345753384112*w^2 - 17057854345563835128788883108*w - 34299957341643443052026541793), hh = 1/155777407310185110629223928*(-12657327167462518499328*w^5 - 18495495027938281568656896*w^4 + 1247708735528258120286326976*w^3 - 9694015543269686983981601520*w^2 + 3966652048288737212257203624*w + 7357662702216901913155444543)*x + 1/467332221930555331887671784*(-263143200386260062268416*w^5 - 384520483678913448260359680*w^4 + 25935239216690935277561086464*w^3 - 201243384412403003124237441216*w^2 + 80517482866691594749291161324*w + 150570853786333685914255361549); degree-2 map to an elliptic curve over the field defined by X^6+2925/2*X^5-13931515/144*X^4+555105005/864*X^3+741300755/1152*X^2-59482995065/62208*X-2129491223327/2985984; E: v^2 = 4u^3 - g2 u - g3 with g2 = w, g3 = 1/4205989997374997986989046056*(-272112803100608279040*w^5-395664740768467375261056*w^4+29647589653215861439432320*w^3-477245639866772050871629032*w^2-195733771623786479752709040*w-207843556759852706226835301) [Genus2Torsion/verify_split_certificates.m, cover 1]" }, "isogenous_product": null, "cover_submitted": true, "cover_accepted": true, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 44, "reducible_chi": 11 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Mazur–Tate 1973; Ogg 1973", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 17 (source: \\cite{MazurTate1973,Ogg1973})", "sources": [ "MazurTate1973", "Ogg1973" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "1", "1" ], "h": [ "1", "0", "1", "1" ], "claimed_group": [ 19 ], "claimed_class": "split", "cover": { "field": "X^6+2925/2*X^5-13931515/144*X^4+555105005/864*X^3+741300755/1152*X^2-59482995065/62208*X-2129491223327/2985984", "cremona": "", "cubic": [ "4", "0", "-(w)", "-(1/4205989997374997986989046056*(-272112803100608279040*w^5-395664740768467375261056*w^4+29647589653215861439432320*w^3-477245639866772050871629032*w^2-195733771623786479752709040*w-207843556759852706226835301))" ], "p": [ "1/350499166447916498915753838*(-1047400442066319243264*w^5-1530244557357870304783296*w^4+103639941866199892102086384*w^3-827402670280905916142666448*w^2+497327212158924622832114979*w+825849916525810185225936473)", "1/175249583223958249457876919*(1181047455487212756480*w^5+1725717953028064158570240*w^4-116547640063212079646312544*w^3+913046320206876174333647256*w^2-421538903357027186361600372*w-655021524942587374393280381)", "1/1401996665791665995663015352*(15990842261263127814144*w^5+23367448923609665432475648*w^4-1575054550234548111640212288*w^3+12162386878499496157273057680*w^2-4320129434415791253147032064*w-8711193399793278371297110063)" ], "q": [ "1/1518829721274304828634933298*(62752690556056115767296*w^5+91701242410418949541259520*w^4-6180068925644118438615566976*w^3+47674259218659743345753384112*w^2-17057854345563835128788883108*w-34299957341643443052026541793)", "1/1518829721274304828634933298*(16541103711402026827776*w^5+24166552645798839366764544*w^4-1636558488650062614742432704*w^3+13063575221221494738458433744*w^2-7641246715946420125902252744*w-11882728243714815255723273527)", "1" ], "hh": [ "1/467332221930555331887671784*(-263143200386260062268416*w^5-384520483678913448260359680*w^4+25935239216690935277561086464*w^3-201243384412403003124237441216*w^2+80517482866691594749291161324*w+150570853786333685914255361549)", "1/155777407310185110629223928*(-12657327167462518499328*w^5-18495495027938281568656896*w^4+1247708735528258120286326976*w^3-9694015543269686983981601520*w^2+3966652048288737212257203624*w+7357662702216901913155444543)" ], "note": "degree-2 map to an elliptic curve over the field defined by X^6+2925/2*X^5-13931515/144*X^4+555105005/864*X^3+741300755/1152*X^2-59482995065/62208*X-2129491223327/2985984; E: v^2 = 4u^3 - g2 u - g3 with g2 = w, g3 = 1/4205989997374997986989046056*(-272112803100608279040*w^5-395664740768467375261056*w^4+29647589653215861439432320*w^3-477245639866772050871629032*w^2-195733771623786479752709040*w-207843556759852706226835301) [Genus2Torsion/verify_split_certificates.m, cover 1]" }, "lmfdb_label_given": "169.a.169.1" }, "source": { "kind": "import", "name": "paper-t2-017", "file": "table2.txt", "line": 23, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:36:07 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.960000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/19.b.log", "conductor_log": "data/logs/19.b.conductor.log", "conductor_cputime_seconds": 0.2 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "20.a", "group": [ 20 ], "group_key": "20", "group_bracket": "[20]", "order": 20, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "0", "0", "0", "0", "-1" ], "h": [ "1", "1" ], "f_str": "-x^5", "h_str": "x + 1", "g": [ "1", "2", "1", "0", "0", "-4" ], "g_str": "-4*x^5 + x^2 + 2*x + 1", "g_scale": "1", "discriminant": "806912", "discriminant_digits": 6, "igusa_clebsch": [ "-320", "-320", "-41536", "12910592" ], "g2_invariants": [ "-6400000/197", "-280000/197", "-3900/197" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "0", "1", "0", "0", "0", "-1" ], "h": [ "0", "0", "1", "1" ] }, "minimal_discriminant": "3152", "minimal_discriminant_factored": "2^4 * 197", "conductor": "394", "conductor_factored": "2 * 197", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "394.a.3152.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/394/a/3152/1", "snapshot_label": "" }, "torsion": { "invariants": [ 20 ], "order": 20, "generators": [ { "order": 20, "mumford": [ "x^2 + 1", "x - 1", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 20, "point_count_primes": [ 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101 ], "claimed": [ 20 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 7, "chi": "T^4 - T^3 - 2*T^2 - 7*T + 49", "primes_tried_below": 500, "admissible_primes": 3, "reducible_chi": 1 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 + 2*T^3 + 2*T^2 + 6*T + 9" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 44, "reducible_chi": 15 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 20 (source: \\cite{Leprevost1997})", "sources": [ "Leprevost1997" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "0", "0", "0", "0", "-1" ], "h": [ "1", "1" ], "claimed_group": [ 20 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "394.a.3152.1" }, "source": { "kind": "import", "name": "paper-t1-020", "file": "table1.txt", "line": 26, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:21 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.230000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/20.a.log", "conductor_log": "data/logs/20.a.conductor.log", "conductor_cputime_seconds": 0.16 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "20.b", "group": [ 20 ], "group_key": "20", "group_bracket": "[20]", "order": 20, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-2", "-7", "0", "3", "-2", "1" ], "h": [ "0", "1", "0", "1" ], "f_str": "x^5 - 2*x^4 + 3*x^3 - 7*x - 2", "h_str": "x^3 + x", "g": [ "-8", "-28", "1", "12", "-6", "4", "1" ], "g_str": "x^6 + 4*x^5 - 6*x^4 + 12*x^3 + x^2 - 28*x - 8", "g_scale": "1", "discriminant": "-28974461227008", "discriminant_digits": 14, "igusa_clebsch": [ "-1600", "-3935168", "56494528", "-28974461227008" ], "g2_invariants": [ "320000000000/7073843073", "31024000000/643076643", "-714280000/58461513" ], "rational_weierstrass_points": 1, "minimal_model": { "f": [ "-2", "-7", "0", "3", "-2", "1" ], "h": [ "0", "1", "0", "1" ] }, "minimal_discriminant": "-7073843073", "minimal_discriminant_factored": "-1 * 3 * 11^9", "conductor": "363", "conductor_factored": "3 * 11^2", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "363.a.3", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-2%2C-7%2C0%2C3%2C-2%2C1%5D%2C%5B0%2C1%2C0%2C1%5D%5D", "snapshot_label": "363.a.3" }, "torsion": { "invariants": [ 20 ], "order": 20, "generators": [ { "order": 20, "mumford": [ "x + 1", "-x^3 - 1", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 40, "point_count_primes": [ 5, 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 20 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "cover", "over": "Q", "detail": "a nonconstant map of degree 3 from y^2 = g to the genus-1 curve v^2 = (1)u^3 + (0)u^2 + (-13392)u + (-1080432) over Q (the elliptic curve 11a1), (x,y) -> (p/q, y*hh/q^2) with p = 564*x^3 - 1536*x^2 + 2100*x - 156, q = x^3 + 5*x^2 - 4*x - 8, hh = 13068*x^3 - 52272*x^2 + 52272*x - 26136; degree-3 map to the elliptic curve 11a1 over Q [Genus2Torsion/verify_split_certificates.m, cover 7]" }, "isogenous_product": null, "cover_submitted": true, "cover_accepted": true, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 18 (source: \\cite{HLP2000})", "sources": [ "HLP2000" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-2", "-7", "0", "3", "-2", "1" ], "h": [ "0", "1", "0", "1" ], "claimed_group": [ 20 ], "claimed_class": "split", "cover": { "field": "", "cremona": "11a1", "cubic": [], "p": [ "-156", "2100", "-1536", "564" ], "q": [ "-8", "-4", "5", "1" ], "hh": [ "-26136", "52272", "-52272", "13068" ], "note": "degree-3 map to the elliptic curve 11a1 over Q [Genus2Torsion/verify_split_certificates.m, cover 7]" }, "lmfdb_label_given": "363.a.3" }, "source": { "kind": "import", "name": "paper-t2-018", "file": "table2.txt", "line": 24, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:36:08 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.21000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/20.b.log", "conductor_log": "data/logs/20.b.conductor.log", "conductor_cputime_seconds": 0.14 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "21.a", "group": [ 21 ], "group_key": "21", "group_bracket": "[21]", "order": 21, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "1", "2", "0", "-1" ], "h": [ "1", "1", "0", "1" ], "f_str": "-x^4 + 2*x^2 + x", "h_str": "x^3 + x + 1", "g": [ "1", "6", "9", "2", "-2", "0", "1" ], "g_str": "x^6 - 2*x^4 + 2*x^3 + 9*x^2 + 6*x + 1", "g_scale": "1", "discriminant": "3178496", "discriminant_digits": 7, "igusa_clebsch": [ "72", "6276", "-109944", "3178496" ], "g2_invariants": [ "59049/776", "-22599/388", "7209/194" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "0", "0", "-2", "1", "0", "-1" ], "h": [ "1", "0", "1", "1" ] }, "minimal_discriminant": "776", "minimal_discriminant_factored": "2^3 * 97", "conductor": "388", "conductor_factored": "2^2 * 97", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "388.a.776.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/388/a/776/1", "snapshot_label": "" }, "torsion": { "invariants": [ 21 ], "order": 21, "generators": [ { "order": 21, "mumford": [ "x", "-x^3 - 1", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 21, "point_count_primes": [ 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 101, 103 ], "claimed": [ 21 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 5, "chi": "T^4 + 2*T^3 + 4*T^2 + 10*T + 25", "primes_tried_below": 500, "admissible_primes": 2, "reducible_chi": 1 }, "q_simple": { "certified": true, "prime": 5, "chi": "T^4 + 2*T^3 + 4*T^2 + 10*T + 25" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 44, "reducible_chi": 15 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 21 (source: \\cite{Leprevost1991})", "sources": [ "Leprevost1991" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "1", "2", "0", "-1" ], "h": [ "1", "1", "0", "1" ], "claimed_group": [ 21 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "388.a.776.1" }, "source": { "kind": "import", "name": "paper-t1-021", "file": "table1.txt", "line": 27, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:21 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.290000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/21.a.log", "conductor_log": "data/logs/21.a.conductor.log", "conductor_cputime_seconds": 0.19 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "21.b", "group": [ 21 ], "group_key": "21", "group_bracket": "[21]", "order": 21, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "0", "0", "1", "2", "2", "1" ], "h": [ "1", "1", "0", "1" ], "f_str": "x^5 + 2*x^4 + 2*x^3 + x^2", "h_str": "x^3 + x + 1", "g": [ "1", "2", "5", "10", "10", "4", "1" ], "g_str": "x^6 + 4*x^5 + 10*x^4 + 10*x^3 + 5*x^2 + 2*x + 1", "g_scale": "1", "discriminant": "-2654208", "discriminant_digits": 7, "igusa_clebsch": [ "-120", "3780", "-18360", "-2654208" ], "g2_invariants": [ "9375/8", "-625/4", "875/18" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "0", "0", "1", "2", "2", "1" ], "h": [ "1", "1", "0", "1" ] }, "minimal_discriminant": "-648", "minimal_discriminant_factored": "-1 * 2^3 * 3^4", "conductor": "324", "conductor_factored": "2^2 * 3^4", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "324.a.648.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/324/a/648/1", "snapshot_label": "" }, "torsion": { "invariants": [ 21 ], "order": 21, "generators": [ { "order": 21, "mumford": [ "x^2 + x", "-2*x - 1", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 21, "point_count_primes": [ 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 21 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 93, "reducible_chi": 34 }, "q_simple": { "certified": true, "prime": 5, "chi": "T^4 - 5*T^2 + 25" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "cover", "over": "X^3 - 5175/4*X^2 + 50625/8*X - 421875/64", "detail": "a nonconstant map of degree 2 from y^2 = g to the genus-1 curve v^2 = (4)u^3 + (0)u^2 + (-w)u + (1/795000*(4192*w^2 + 353700*w - 245625)) over X^3 - 5175/4*X^2 + 50625/8*X - 421875/64, (x,y) -> (p/q, y*hh/q^2) with p = 1/397500*(-16*w^2 + 64900*w + 50625)*x^2 + 1/99375*(24*w^2 - 15200*w + 73125)*x + 1/397500*(64*w^2 + 16000*w + 95625), q = x^2 + 1/298125*(256*w^2 - 328200*w + 978750)*x + 1/59625*(16*w^2 - 19320*w + 68625), hh = 1/33125*(-64*w^2 - 5400*w + 3750)*x + 1/19875*(-224*w^2 + 3360*w - 6750); degree-2 map to an elliptic curve over the field defined by X^3-5175/4*X^2+50625/8*X-421875/64; E: v^2 = 4u^3 - g2 u - g3 with g2 = w, g3 = 1/795000*(-4192*w^2-353700*w+245625) [Genus2Torsion/verify_split_certificates.m, cover 2]" }, "isogenous_product": null, "cover_submitted": true, "cover_accepted": true, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 44, "reducible_chi": 20 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Ogg 1973", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 19 (source: \\cite{Ogg1973})", "sources": [ "Ogg1973" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "0", "1", "2", "2", "1" ], "h": [ "1", "1", "0", "1" ], "claimed_group": [ 21 ], "claimed_class": "split", "cover": { "field": "X^3-5175/4*X^2+50625/8*X-421875/64", "cremona": "", "cubic": [ "4", "0", "-(w)", "-(1/795000*(-4192*w^2-353700*w+245625))" ], "p": [ "1/397500*(64*w^2+16000*w+95625)", "1/99375*(24*w^2-15200*w+73125)", "1/397500*(-16*w^2+64900*w+50625)" ], "q": [ "1/59625*(16*w^2-19320*w+68625)", "1/298125*(256*w^2-328200*w+978750)", "1" ], "hh": [ "1/19875*(-224*w^2+3360*w-6750)", "1/33125*(-64*w^2-5400*w+3750)" ], "note": "degree-2 map to an elliptic curve over the field defined by X^3-5175/4*X^2+50625/8*X-421875/64; E: v^2 = 4u^3 - g2 u - g3 with g2 = w, g3 = 1/795000*(-4192*w^2-353700*w+245625) [Genus2Torsion/verify_split_certificates.m, cover 2]" }, "lmfdb_label_given": "324.a.648.1" }, "source": { "kind": "import", "name": "paper-t2-019", "file": "table2.txt", "line": 25, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:36:10 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.10000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/21.b.log", "conductor_log": "data/logs/21.b.conductor.log", "conductor_cputime_seconds": 0.21 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "21.c", "group": [ 21 ], "group_key": "21", "group_bracket": "[21]", "order": 21, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "1", "3", "3", "1" ], "h": [ "0", "1", "1", "1" ], "f_str": "x^3 + 3*x^2 + 3*x + 1", "h_str": "x^3 + x^2 + x", "g": [ "4", "12", "13", "6", "3", "2", "1" ], "g_str": "x^6 + 2*x^5 + 3*x^4 + 6*x^3 + 13*x^2 + 12*x + 4", "g_scale": "1", "discriminant": "-22151168", "discriminant_digits": 8, "igusa_clebsch": [ "-408", "13092", "-1289688", "-22151168" ], "g2_invariants": [ "345025251/5408", "-928557/1352", "0" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "0", "1", "2", "1", "2", "1" ], "h": [ "1", "0", "0", "1" ] }, "minimal_discriminant": "-5408", "minimal_discriminant_factored": "-1 * 2^5 * 13^2", "conductor": "676", "conductor_factored": "26 * 26 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 26 * 26 for the elliptic curves ['1', '0', '1', '0', '0'] and ['1', '-1', '1', '-3', '3'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "676" }, "lmfdb": { "label": "676.a.5408.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/676/a/5408/1", "snapshot_label": "" }, "torsion": { "invariants": [ 21 ], "order": 21, "generators": [ { "order": 21, "mumford": [ "1", "x^3 + x^2", "2" ] } ], "method": "generators", "generators_note": "", "generators_submitted": 1, "point_count_gcd": 21, "point_count_primes": [ 3, 5, 7, 11, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 21 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 93, "reducible_chi": 93 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x : y : -x - z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "0", "1", "0", "0" ], [ "1", "-1", "1", "-3", "3" ] ], "conductors": [ "26", "26" ], "conductor": "676", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 44, "reducible_chi": 44 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "F. Leprévost", "year": 1995, "historical": false }, "reference": "F. Leprévost, Jacobiennes de certaines courbes de genre 2: torsion et simplicité, J. Théor. Nombres Bordeaux 7 (1995), 283–306, https://doi.org/10.5802/jtnb.144: the curve C21 of §2.1 (Théorème 1.2.1), y^2 = 4x^6 - 12x^5 + 13x^4 - 6x^3 + 3x^2 - 2x + 1, shown in §3.2 to be isogenous over Q to E1 x E2 (points of order 3 and 7). This model is LMFDB 676.a.5408.1, https://www.lmfdb.org/Genus2Curve/Q/676/a/5408/1", "sources": [], "rm": false, "notes": "Q-split witness: the census has only geometrically simple and Q-simple (geometrically split) curves with this torsion group. Not a new curve. LMFDB 676.a.5408.1 (torsion [21], is_simple_base = false) is isomorphic over Q to Leprévost's C21 (Magma IsIsomorphic; equal G2-invariants [345025251/5408, -928557/1352, 0]), so the curve is credited to Leprévost 1995. Output of pipeline/magma/prepare_submission.m (Magma V2.29-4): J(Q)_tors = [21]; gcd of #J(F_p) over 25 good primes = 21 (sharp); no strict prime below 500; conductor 676 = 2^2 * 13^2.", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "1", "3", "3", "1" ], "h": [ "0", "1", "1", "1" ], "claimed_group": [ 21 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "issue", "number": 7, "url": "https://github.com/Genus-2-torsion/Genus-2-torsion.github.io/issues/7" }, "dates": { "submitted": "2026-10-14", "verified": "2026-10-14 10:49:22 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.36000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/21.c.log", "conductor_log": "data/logs/21.c.conductor.log", "conductor_cputime_seconds": 0.23 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "22.a", "group": [ 22 ], "group_key": "22", "group_bracket": "[22]", "order": 22, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "1", "-1", "-2", "1" ], "h": [ "0", "1", "0", "1" ], "f_str": "x^3 - 2*x^2 - x + 1", "h_str": "x^3 + x", "g": [ "4", "-4", "-7", "4", "2", "0", "1" ], "g_str": "x^6 + 2*x^4 + 4*x^3 - 7*x^2 - 4*x + 4", "g_scale": "1", "discriminant": "-78118912", "discriminant_digits": 8, "igusa_clebsch": [ "-640", "50944", "-17393920", "-78118912" ], "g2_invariants": [ "25600000/149", "-1056000/149", "-861000/149" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "1", "-1", "-2", "1" ], "h": [ "0", "1", "0", "1" ] }, "minimal_discriminant": "-19072", "minimal_discriminant_factored": "-1 * 2^7 * 149", "conductor": "1192", "conductor_factored": "2^3 * 149", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "1192.a.19072.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/1192/a/19072/1", "snapshot_label": "" }, "torsion": { "invariants": [ 22 ], "order": 22, "generators": [ { "order": 22, "mumford": [ "x", "-x^3 + 2", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 22, "point_count_primes": [ 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101 ], "claimed": [ 22 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 3, "chi": "T^4 + 2*T^3 + 4*T^2 + 6*T + 9", "primes_tried_below": 500, "admissible_primes": 1, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 + 2*T^3 + 4*T^2 + 6*T + 9" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 44, "reducible_chi": 11 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 22 (source: \\cite{Leprevost1993})", "sources": [ "Leprevost1993" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "1", "-1", "-2", "1" ], "h": [ "0", "1", "0", "1" ], "claimed_group": [ 22 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "1192.a.19072.1" }, "source": { "kind": "import", "name": "paper-t1-022", "file": "table1.txt", "line": 28, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:22 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.260000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/22.a.log", "conductor_log": "data/logs/22.a.conductor.log", "conductor_cputime_seconds": 0.15 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "23.a", "group": [ 23 ], "group_key": "23", "group_bracket": "[23]", "order": 23, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "1", "-2", "2", "1", "-2", "-1", "1" ], "h": [ "0", "0", "1" ], "f_str": "x^6 - x^5 - 2*x^4 + x^3 + 2*x^2 - 2*x + 1", "h_str": "x^2", "g": [ "4", "-8", "8", "4", "-7", "-4", "4" ], "g_str": "4*x^6 - 4*x^5 - 7*x^4 + 4*x^3 + 8*x^2 - 8*x + 4", "g_scale": "1", "discriminant": "-7113539584", "discriminant_digits": 10, "igusa_clebsch": [ "-1568", "397888", "-235899392", "-7113539584" ], "g2_invariants": [ "282475249/1696", "-352947/32", "-6245001/1696" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "1", "-2", "2", "1", "-2", "-1", "1" ], "h": [ "0", "0", "1" ] }, "minimal_discriminant": "-1736704", "minimal_discriminant_factored": "-1 * 2^15 * 53", "conductor": "1696", "conductor_factored": "2^5 * 53", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "1696.c.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B1%2C-2%2C2%2C1%2C-2%2C-1%2C1%5D%2C%5B0%2C0%2C1%5D%5D", "snapshot_label": "1696.c.1" }, "torsion": { "invariants": [ 23 ], "order": 23, "generators": [ { "order": 23, "mumford": [ "x^2 - 1/2*x - 1", "2*x - 2", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 23, "point_count_primes": [ 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 23 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 3, "chi": "T^4 + 2*T^3 + 5*T^2 + 6*T + 9", "primes_tried_below": 500, "admissible_primes": 1, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 + 2*T^3 + 5*T^2 + 6*T + 9" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 44, "reducible_chi": 5 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Leprévost 1993", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 23 (source: \\cite{Leprevost1993})", "sources": [ "Leprevost1993" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "1", "-2", "2", "1", "-2", "-1", "1" ], "h": [ "0", "0", "1" ], "claimed_group": [ 23 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "1696.c.1" }, "source": { "kind": "import", "name": "paper-t1-023", "file": "table1.txt", "line": 29, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:22 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.270000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/23.a.log", "conductor_log": "data/logs/23.a.conductor.log", "conductor_cputime_seconds": 0.17 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "24.a", "group": [ 24 ], "group_key": "24", "group_bracket": "[24]", "order": 24, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "2", "4", "3", "2" ], "h": [ "1", "0", "0", "1" ], "f_str": "2*x^4 + 3*x^3 + 4*x^2 + 2*x", "h_str": "x^3 + 1", "g": [ "1", "8", "16", "14", "8", "0", "1" ], "g_str": "x^6 + 8*x^4 + 14*x^3 + 16*x^2 + 8*x + 1", "g_scale": "1", "discriminant": "750256128", "discriminant_digits": 9, "igusa_clebsch": [ "-1112", "32356", "-11159576", "750256128" ], "g2_invariants": [ "-51888844699/183168", "-34913047/5088", "19321/1272" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "0", "0", "2", "-4", "4", "-2" ], "h": [ "1", "0", "0", "1" ] }, "minimal_discriminant": "183168", "minimal_discriminant_factored": "2^7 * 3^3 * 53", "conductor": "1908", "conductor_factored": "2^2 * 3^2 * 53", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "1908.a.183168.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/1908/a/183168/1", "snapshot_label": "" }, "torsion": { "invariants": [ 24 ], "order": 24, "generators": [ { "order": 24, "mumford": [ "x^2 + 2*x + 1", "-5*x - 3", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 48, "point_count_primes": [ 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 24 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 17, "chi": "T^4 + T^3 - 20*T^2 + 17*T + 289", "primes_tried_below": 500, "admissible_primes": 5, "reducible_chi": 3 }, "q_simple": { "certified": true, "prime": 5, "chi": "T^4 + 3*T^3 + 4*T^2 + 15*T + 25" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 43, "reducible_chi": 16 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 24 (source: \\cite{Leprevost1993})", "sources": [ "Leprevost1993" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "2", "4", "3", "2" ], "h": [ "1", "0", "0", "1" ], "claimed_group": [ 24 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "1908.a.183168.1" }, "source": { "kind": "import", "name": "paper-t1-024", "file": "table1.txt", "line": 30, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:23 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.310000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/24.a.log", "conductor_log": "data/logs/24.a.conductor.log", "conductor_cputime_seconds": 0.31 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "24.b", "group": [ 24 ], "group_key": "24", "group_bracket": "[24]", "order": 24, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "375", "0", "97", "0", "8" ], "h": [ "0", "1", "0", "1" ], "f_str": "8*x^4 + 97*x^2 + 375", "h_str": "x^3 + x", "g": [ "1500", "0", "389", "0", "34", "0", "1" ], "g_str": "x^6 + 34*x^4 + 389*x^2 + 1500", "g_scale": "1", "discriminant": "-15360000000", "discriminant_digits": 11, "igusa_clebsch": [ "-571616", "1206016", "-229542955264", "-15360000000" ], "g2_invariants": [ "116399727907379248994752/234375", "193988089667923989064/9375", "28735596382851136/25" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "375", "0", "97", "0", "8" ], "h": [ "0", "1", "0", "1" ] }, "minimal_discriminant": "-3750000", "minimal_discriminant_factored": "-1 * 2^4 * 3 * 5^7", "conductor": "300", "conductor_factored": "15 * 20 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 15 * 20 for the elliptic curves ['1', '1', '1', '35', '-28'] and ['0', '1', '0', '4', '4'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "300" }, "lmfdb": { "label": "300.a.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B375%2C0%2C97%2C0%2C8%5D%2C%5B0%2C1%2C0%2C1%5D%5D", "snapshot_label": "300.a.1" }, "torsion": { "invariants": [ 24 ], "order": 24, "generators": [ { "order": 24, "mumford": [ "x^2 - 1/3*x + 34/3", "2/9*x + 16/9", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 48, "point_count_primes": [ 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 24 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x : y : -z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "1", "1", "35", "-28" ], [ "0", "1", "0", "4", "4" ] ], "conductors": [ "15", "20" ], "conductor": "300", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [ "16.b" ], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 20 (source: \\cite{Leprevost1995})", "sources": [ "Leprevost1995" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "375", "0", "97", "0", "8" ], "h": [ "0", "1", "0", "1" ], "claimed_group": [ 24 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "300.a.1" }, "source": { "kind": "import", "name": "paper-t2-020", "file": "table2.txt", "line": 26, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:36:11 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.16000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/24.b.log", "conductor_log": "data/logs/24.b.conductor.log", "conductor_cputime_seconds": 0.19 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "25.a", "group": [ 25 ], "group_key": "25", "group_bracket": "[25]", "order": 25, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "2", "-23", "41", "75", "25", "-9" ], "h": [ "1", "1" ], "f_str": "-9*x^5 + 25*x^4 + 75*x^3 + 41*x^2 - 23*x + 2", "h_str": "x + 1", "g": [ "9", "-90", "165", "300", "100", "-36" ], "g_str": "-36*x^5 + 100*x^4 + 300*x^3 + 165*x^2 - 90*x + 9", "g_scale": "1", "discriminant": "-2834352000000000", "discriminant_digits": 16, "igusa_clebsch": [ "405600", "20487960000", "1662601599000000", "-3673320192000000000" ], "g2_invariants": [ "-22057358695840/59049", "33787663000/2187", "-586214525/486" ], "rational_weierstrass_points": 1, "minimal_model": { "f": [ "2", "-23", "41", "75", "25", "-9" ], "h": [ "1", "1" ] }, "minimal_discriminant": "-896806687500000", "minimal_discriminant_factored": "-1 * 2^5 * 3^15 * 5^9", "conductor": "168750", "conductor_factored": "2 * 3^3 * 5^5", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "168750.e.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B2%2C-23%2C41%2C75%2C25%2C-9%5D%2C%5B1%2C1%5D%5D", "snapshot_label": "168750.e.2" }, "torsion": { "invariants": [ 25 ], "order": 25, "generators": [ { "order": 25, "mumford": [ "x^2 - 14/3*x + 1", "55/3*x - 5", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 25, "point_count_primes": [ 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 25 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 11, "chi": "T^4 - 5*T^3 + 13*T^2 - 55*T + 121", "primes_tried_below": 500, "admissible_primes": 2, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 7, "chi": "T^4 + 49" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 43, "reducible_chi": 10 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Nicholls 2018", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 25 (source: \\cite{Nicholls2018})", "sources": [ "Nicholls2018" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "2", "-23", "41", "75", "25", "-9" ], "h": [ "1", "1" ], "claimed_group": [ 25 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "168750.e.2" }, "source": { "kind": "import", "name": "paper-t1-025", "file": "table1.txt", "line": 31, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:31 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.260000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/25.a.log", "conductor_log": "data/logs/25.a.conductor.log", "conductor_cputime_seconds": 7.56 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "25.b", "group": [ 25 ], "group_key": "25", "group_bracket": "[25]", "order": 25, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "1", "8", "13", "9", "14", "4", "2" ], "h": [ "0", "0", "1", "1" ], "f_str": "2*x^6 + 4*x^5 + 14*x^4 + 9*x^3 + 13*x^2 + 8*x + 1", "h_str": "x^3 + x^2", "g": [ "4", "32", "52", "36", "57", "18", "9" ], "g_str": "9*x^6 + 18*x^5 + 57*x^4 + 36*x^3 + 52*x^2 + 32*x + 4", "g_scale": "1", "discriminant": "13980445502865408", "discriminant_digits": 17, "igusa_clebsch": [ "-25248", "79073856", "-495118600704", "13980445502865408" ], "g2_invariants": [ "-1258284197543/13716864", "8604554431/2286144", "215876449/1143072" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "2", "4", "14", "9", "13", "8", "1" ], "h": [ "1", "1" ] }, "minimal_discriminant": "3413194702848", "minimal_discriminant_factored": "2^17 * 3^12 * 7^2", "conductor": "42336", "conductor_factored": "2^5 * 3^3 * 7^2", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "42336.dw.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B1%2C8%2C13%2C9%2C14%2C4%2C2%5D%2C%5B0%2C0%2C1%2C1%5D%5D", "snapshot_label": "42336.dw.1" }, "torsion": { "invariants": [ 25 ], "order": 25, "generators": [ { "order": 25, "mumford": [ "x^2 - 2/3*x - 2/3", "32/3*x + 20/3", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 25, "point_count_primes": [ 5, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 25 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 51 }, "q_simple": { "certified": true, "prime": 5, "chi": "T^4 - T^2 + 25" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "cover", "over": "X^2 - 32246250*X + 2573441015625", "detail": "a nonconstant map of degree 5 from y^2 = g to the genus-1 curve v^2 = (4)u^3 + (0)u^2 + (-w)u + (1/52*(-113515*w + 9308159375)) over X^2 - 32246250*X + 2573441015625, (x,y) -> (p/q, y*hh/q^2) with p = 1/39000*(17*w + 1831875)*x^5 + 1/17784*(5*w - 587625)*x^4 + 1/85500*(17*w + 6631875)*x^3 + 1/18525*(4*w + 499375)*x^2 + 1/7125*(w + 264375)*x + 1/92625*(2*w + 4803750), q = x^5 + 1/13893750*(w - 2229375)*x^4 + 1/6946875*(w + 7033125)*x^3 + 1/13893750*(w + 2401875)*x^2, hh = 1/325*(11*w - 879375)*x^7 + 1/18525*(601*w - 47735625)*x^6 + 1/18525*(23*w - 331875)*x^5 + 1/55575*(-709*w + 66208125)*x^4 + 1/55575*(-808*w + 78510000)*x^3 + 1/2223*(-14*w + 1200750)*x^2 + 1/55575*(-56*w + 13695000)*x; degree-5 map to an elliptic curve over the field defined by X^2-32246250*X+2573441015625; E: v^2 = 4u^3 - g2 u - g3 with g2 = w, g3 = 1/52*(113515*w-9308159375) [Genus2Torsion/verify_split_certificates.m, cover 11]" }, "isogenous_product": null, "cover_submitted": true, "cover_accepted": true, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 23 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Leprévost 1995; Platonov–Zhgun–Petrunin 2013", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 21 (source: \\cite{Leprevost1995,PZP2013})", "sources": [ "Leprevost1995", "PZP2013" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "1", "8", "13", "9", "14", "4", "2" ], "h": [ "0", "0", "1", "1" ], "claimed_group": [ 25 ], "claimed_class": "split", "cover": { "field": "X^2-32246250*X+2573441015625", "cremona": "", "cubic": [ "4", "0", "-(w)", "-(1/52*(113515*w-9308159375))" ], "p": [ "1/92625*(2*w+4803750)", "1/7125*(w+264375)", "1/18525*(4*w+499375)", "1/85500*(17*w+6631875)", "1/17784*(5*w-587625)", "1/39000*(17*w+1831875)" ], "q": [ "0", "0", "1/13893750*(w+2401875)", "1/6946875*(w+7033125)", "1/13893750*(w-2229375)", "1" ], "hh": [ "0", "1/55575*(-56*w+13695000)", "1/2223*(-14*w+1200750)", "1/55575*(-808*w+78510000)", "1/55575*(-709*w+66208125)", "1/18525*(23*w-331875)", "1/18525*(601*w-47735625)", "1/325*(11*w-879375)" ], "note": "degree-5 map to an elliptic curve over the field defined by X^2-32246250*X+2573441015625; E: v^2 = 4u^3 - g2 u - g3 with g2 = w, g3 = 1/52*(113515*w-9308159375) [Genus2Torsion/verify_split_certificates.m, cover 11]" }, "lmfdb_label_given": "42336.dw.1" }, "source": { "kind": "import", "name": "paper-t2-021", "file": "table2.txt", "line": 27, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:36:19 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.880000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/25.b.log", "conductor_log": "data/logs/25.b.conductor.log", "conductor_cputime_seconds": 7.12 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "26.a", "group": [ 26 ], "group_key": "26", "group_bracket": "[26]", "order": 26, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "0", "-3", "-3", "-3", "-12" ], "h": [ "1", "1" ], "f_str": "-12*x^5 - 3*x^4 - 3*x^3 - 3*x^2", "h_str": "x + 1", "g": [ "1", "2", "-11", "-12", "-12", "-48" ], "g_str": "-48*x^5 - 12*x^4 - 12*x^3 - 11*x^2 + 2*x + 1", "g_scale": "1", "discriminant": "-98680799232", "discriminant_digits": 11, "igusa_clebsch": [ "-5088", "-8452800", "6011103744", "-227360561430528" ], "g2_invariants": [ "418195493/223074", "216913789/446148", "-88250353/2676888" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "0", "-12", "-3", "-3", "-3" ], "h": [ "0", "0", "1", "1" ] }, "minimal_discriminant": "-55507949568", "minimal_discriminant_factored": "-1 * 2^11 * 3^13 * 17", "conductor": "11016", "conductor_factored": "2^3 * 3^4 * 17", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "11016.g.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C0%2C-3%2C-3%2C-3%2C-12%5D%2C%5B1%2C1%5D%5D", "snapshot_label": "11016.g.1" }, "torsion": { "invariants": [ 26 ], "order": 26, "generators": [ { "order": 26, "mumford": [ "x^2 + x", "-5*x + 1", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 26, "point_count_primes": [ 5, 7, 11, 13, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 26 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 11, "chi": "T^4 + 4*T^3 + 12*T^2 + 44*T + 121", "primes_tried_below": 500, "admissible_primes": 3, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 5, "chi": "T^4 + 25" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 43, "reducible_chi": 10 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 26 (source: \\cite{Leprevost1993})", "sources": [ "Leprevost1993" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "0", "-3", "-3", "-3", "-12" ], "h": [ "1", "1" ], "claimed_group": [ 26 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "11016.g.1" }, "source": { "kind": "import", "name": "paper-t1-026", "file": "table1.txt", "line": 32, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:32 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.250000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/26.a.log", "conductor_log": "data/logs/26.a.conductor.log", "conductor_cputime_seconds": 1.01 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "27.a", "group": [ 27 ], "group_key": "27", "group_bracket": "[27]", "order": 27, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "-1", "1", "1", "-1" ], "h": [ "1", "0", "0", "1" ], "f_str": "-x^4 + x^3 + x^2 - x", "h_str": "x^3 + 1", "g": [ "1", "-4", "4", "6", "-4", "0", "1" ], "g_str": "x^6 - 4*x^4 + 6*x^3 + 4*x^2 - 4*x + 1", "g_scale": "1", "discriminant": "39583744", "discriminant_digits": 8, "igusa_clebsch": [ "232", "25060", "762216", "39583744" ], "g2_invariants": [ "20511149/9664", "-2755957/4832", "175769/2416" ], "rational_weierstrass_points": 1, "minimal_model": { "f": [ "0", "-1", "1", "1", "-1" ], "h": [ "1", "0", "0", "1" ] }, "minimal_discriminant": "9664", "minimal_discriminant_factored": "2^6 * 151", "conductor": "604", "conductor_factored": "2^2 * 151", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "604.a.9664.2", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/604/a/9664/2", "snapshot_label": "" }, "torsion": { "invariants": [ 27 ], "order": 27, "generators": [ { "order": 27, "mumford": [ "x", "x^3 - 1", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 27, "point_count_primes": [ 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101 ], "claimed": [ 27 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 3, "chi": "T^4 + 3*T^3 + 5*T^2 + 9*T + 9", "primes_tried_below": 500, "admissible_primes": 1, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 + 3*T^3 + 5*T^2 + 9*T + 9" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 44, "reducible_chi": 8 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Howe 2015", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 27 (source: \\cite{Howe2015})", "sources": [ "Howe2015" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "-1", "1", "1", "-1" ], "h": [ "1", "0", "0", "1" ], "claimed_group": [ 27 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "604.a.9664.2" }, "source": { "kind": "import", "name": "paper-t1-027", "file": "table1.txt", "line": 33, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:33 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.320000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/27.a.log", "conductor_log": "data/logs/27.a.conductor.log", "conductor_cputime_seconds": 0.18 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "27.b", "group": [ 27 ], "group_key": "27", "group_bracket": "[27]", "order": 27, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "0", "4", "20", "17", "9", "0", "1" ], "h": [ "1", "1" ], "f_str": "x^6 + 9*x^4 + 17*x^3 + 20*x^2 + 4*x", "h_str": "x + 1", "g": [ "1", "18", "81", "68", "36", "0", "4" ], "g_str": "4*x^6 + 36*x^4 + 68*x^3 + 81*x^2 + 18*x + 1", "g_scale": "1", "discriminant": "1141260857376768", "discriminant_digits": 16, "igusa_clebsch": [ "-19872", "62005824", "-363566320128", "1141260857376768" ], "g2_invariants": [ "-173781261/512", "2737575/128", "106329/32" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "0", "4", "20", "17", "9", "0", "1" ], "h": [ "1", "1" ] }, "minimal_discriminant": "278628139008", "minimal_discriminant_factored": "2^19 * 3^12", "conductor": "7776", "conductor_factored": "2^5 * 3^5", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "7776.e.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C4%2C20%2C17%2C9%2C0%2C1%5D%2C%5B1%2C1%5D%5D", "snapshot_label": "7776.e.2" }, "torsion": { "invariants": [ 27 ], "order": 27, "generators": [ { "order": 27, "mumford": [ "x^2 - 5/2*x - 1/2", "51/2*x + 9/2", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 27, "point_count_primes": [ 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 27 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 93, "reducible_chi": 55 }, "q_simple": { "certified": true, "prime": 7, "chi": "T^4 + 4*T^2 + 49" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "cover", "over": "X^2 + 138024*X - 1767690864", "detail": "a nonconstant map of degree 3 from y^2 = g to the genus-1 curve v^2 = (4)u^3 + (0)u^2 + (-w)u + (-93*w + 1344276) over X^2 + 138024*X - 1767690864, (x,y) -> (p/q, y*hh/q^2) with p = 1/648*(-w - 4860)*x^3 + 1/5184*(13*w - 1342332)*x^2 + 1/10368*(7*w + 1042956)*x + 1/10368*(-5*w - 438372), q = x^3 + 1/46656*(-w - 69012)*x^2 + 1/93312*(5*w + 904932)*x + 1/93312*(w + 162324), hh = 1/36*(-w + 24300)*x^3 + 1/72*(w - 70956)*x^2 + 1/72*(-w - 208980)*x + 1/72*(-w - 162324); degree-3 map to an elliptic curve over the field defined by X^2+138024*X-1767690864; E: v^2 = 4u^3 - g2 u - g3 with g2 = w, g3 = 93*w-1344276 [Genus2Torsion/verify_split_certificates.m, cover 12]" }, "isogenous_product": null, "cover_submitted": true, "cover_accepted": true, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 44, "reducible_chi": 26 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Howe 2015", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 22 (source: \\cite{Leprevost1995})", "sources": [ "Howe2015" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "4", "20", "17", "9", "0", "1" ], "h": [ "1", "1" ], "claimed_group": [ 27 ], "claimed_class": "split", "cover": { "field": "X^2+138024*X-1767690864", "cremona": "", "cubic": [ "4", "0", "-(w)", "-(93*w-1344276)" ], "p": [ "1/10368*(-5*w-438372)", "1/10368*(7*w+1042956)", "1/5184*(13*w-1342332)", "1/648*(-w-4860)" ], "q": [ "1/93312*(w+162324)", "1/93312*(5*w+904932)", "1/46656*(-w-69012)", "1" ], "hh": [ "1/72*(-w-162324)", "1/72*(-w-208980)", "1/72*(w-70956)", "1/36*(-w+24300)" ], "note": "degree-3 map to an elliptic curve over the field defined by X^2+138024*X-1767690864; E: v^2 = 4u^3 - g2 u - g3 with g2 = w, g3 = 93*w-1344276 [Genus2Torsion/verify_split_certificates.m, cover 12]" }, "lmfdb_label_given": "7776.e.2" }, "source": { "kind": "import", "name": "paper-t2-022", "file": "table2.txt", "line": 28, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:37:17 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.20000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/27.b.log", "conductor_log": "data/logs/27.b.conductor.log", "conductor_cputime_seconds": 56.52 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "27.c", "group": [ 27 ], "group_key": "27", "group_bracket": "[27]", "order": 27, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "9", "-36", "90", "-192", "249", "-60", "4" ], "h": [ "0" ], "f_str": "4*x^6 - 60*x^5 + 249*x^4 - 192*x^3 + 90*x^2 - 36*x + 9", "h_str": "0", "g": [ "36", "-144", "360", "-768", "996", "-240", "16" ], "g_str": "16*x^6 - 240*x^5 + 996*x^4 - 768*x^3 + 360*x^2 - 144*x + 36", "g_scale": "1", "discriminant": "64621732218373901647872", "discriminant_digits": 23, "igusa_clebsch": [ "-953856", "356705648640", "-98351398854328320", "64621732218373901647872" ], "g2_invariants": [ "-1564031349/1024", "171810207/512", "10094907/256" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "10", "-6", "-4", "11", "-1", "-3", "1" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "14693280768", "minimal_discriminant_factored": "2^10 * 3^15", "conductor": "8748", "conductor_factored": "2^2 * 3^7", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B9%2C-36%2C90%2C-192%2C249%2C-60%2C4%5D%2C%5B0%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 27 ], "order": 27, "generators": [ { "order": 27, "mumford": [ "x", "2*x^3 - 3", "2" ] } ], "method": "generators", "generators_note": "", "generators_submitted": 1, "point_count_gcd": 27, "point_count_primes": [ 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 27 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 93, "reducible_chi": 93 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "cover", "over": "Q", "detail": "a nonconstant map of degree 3 from y^2 = g to the genus-1 curve v^2 = (648)u^3 + (297)u^2 + (18)u + (1) over Q, (x,y) -> (p/q, y*hh/q^2) with p = x^2 - 2*x + 1, q = 2*x^3 - 15*x^2 - 3, hh = x^3 - 3*x^2 + 15*x + 3; Leprévost 1995, §3.2: the degree-3 map X = (x-1)^2/(2x^3-15x^2-3) onto E1: Y^2 = 648X^3 + 297X^2 + 18X + 1 (54b3), written on the even model y^2 = 4f" }, "isogenous_product": null, "cover_submitted": true, "cover_accepted": true, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 44, "reducible_chi": 44 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "F. Leprévost", "year": 1995, "historical": false }, "reference": "F. Leprévost, Jacobiennes de certaines courbes de genre 2: torsion et simplicité, J. Théor. Nombres Bordeaux 7 (1995), 283–306, https://doi.org/10.5802/jtnb.144: the curve C27 of Théorème 1.2.1 (p. 293), y^2 = (2x^3 - 15x^2 - 3)(2x^3 - 15x^2 + 12x - 3); §3.2 (p. 302) gives the explicit degree-3 cover onto E1 and shows that its Jacobian is not simple over Q. Not in the production LMFDB.", "sources": [], "rm": false, "notes": "Q-split witness for [27]: the census has only a geometrically simple curve and a Q-simple geometrically split curve (27.b) with this torsion group. Not a new curve: this is Leprévost's C27 (f above is (2x^3 - 15x^2 - 3)(2x^3 - 15x^2 + 12x - 3) expanded). It is not the census curve 27.b (different G2-invariants). Output of pipeline/magma/prepare_submission.m (Magma V2.29-4): J(Q)_tors = [27]; gcd of #J(F_p) over 25 good primes = 27 (sharp); no strict prime below 500; conductor (Magma, Ogg's formula at 2) 8748 = 2^2 * 3^7.", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "9", "-36", "90", "-192", "249", "-60", "4" ], "h": [ "0" ], "claimed_group": [ 27 ], "claimed_class": "split", "cover": { "field": "", "cremona": "", "cubic": [ "648", "297", "18", "1" ], "p": [ "1", "-2", "1" ], "q": [ "-3", "0", "-15", "2" ], "hh": [ "3", "15", "-3", "1" ], "note": "Leprévost 1995, §3.2: the degree-3 map X = (x-1)^2/(2x^3-15x^2-3) onto E1: Y^2 = 648X^3 + 297X^2 + 18X + 1 (54b3), written on the even model y^2 = 4f" }, "lmfdb_label_given": "" }, "source": { "kind": "issue", "number": 8, "url": "https://github.com/Genus-2-torsion/Genus-2-torsion.github.io/issues/8" }, "dates": { "submitted": "2026-10-14", "verified": "2026-10-14 10:49:32 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.35000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/27.c.log", "conductor_log": "data/logs/27.c.conductor.log", "conductor_cputime_seconds": 4.39 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "28.a", "group": [ 28 ], "group_key": "28", "group_bracket": "[28]", "order": 28, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "2", "3", "1", "1", "0", "-1" ], "h": [ "1", "0", "0", "1" ], "f_str": "-x^5 + x^3 + x^2 + 3*x + 2", "h_str": "x^3 + 1", "g": [ "9", "12", "4", "6", "0", "-4", "1" ], "g_str": "x^6 - 4*x^5 + 6*x^3 + 4*x^2 + 12*x + 9", "g_scale": "1", "discriminant": "-27537408", "discriminant_digits": 8, "igusa_clebsch": [ "-3864", "351588", "-526124568", "-27537408" ], "g2_invariants": [ "324526850403/83", "25281736298/249", "-4178776252/747" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "2", "3", "1", "1", "0", "-1" ], "h": [ "1", "0", "0", "1" ] }, "minimal_discriminant": "-6723", "minimal_discriminant_factored": "-1 * 3^4 * 83", "conductor": "249", "conductor_factored": "3 * 83", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "249.a.6723.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/249/a/6723/1", "snapshot_label": "" }, "torsion": { "invariants": [ 28 ], "order": 28, "generators": [ { "order": 28, "mumford": [ "x", "-x^3 - 3", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 28, "point_count_primes": [ 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 89, 97, 101, 103, 107 ], "claimed": [ 28 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 7, "chi": "T^4 + T^3 - 2*T^2 + 7*T + 49", "primes_tried_below": 500, "admissible_primes": 2, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 5, "chi": "T^4 + 2*T^2 + 25" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 43, "reducible_chi": 12 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Platonov–Petrunin 2012a; Platonov–Zhgun–Petrunin 2013", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 28 (source: \\cite{PP2012,PZP2013})", "sources": [ "PP2012", "PZP2013" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "2", "3", "1", "1", "0", "-1" ], "h": [ "1", "0", "0", "1" ], "claimed_group": [ 28 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "249.a.6723.1" }, "source": { "kind": "import", "name": "paper-t1-028", "file": "table1.txt", "line": 34, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:33 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.310000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/28.a.log", "conductor_log": "data/logs/28.a.conductor.log", "conductor_cputime_seconds": 0.1 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "28.b", "group": [ 28 ], "group_key": "28", "group_bracket": "[28]", "order": 28, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "4", "10", "-4", "-1", "5", "-3", "1" ], "h": [ "0", "1", "1" ], "f_str": "x^6 - 3*x^5 + 5*x^4 - x^3 - 4*x^2 + 10*x + 4", "h_str": "x^2 + x", "g": [ "16", "40", "-15", "-2", "21", "-12", "4" ], "g_str": "4*x^6 - 12*x^5 + 21*x^4 - 2*x^3 - 15*x^2 + 40*x + 16", "g_scale": "1", "discriminant": "528418416361472", "discriminant_digits": 15, "igusa_clebsch": [ "-29496", "50427972", "-371068025400", "528418416361472" ], "g2_invariants": [ "-681342826828707207/129008402432", "-515293627830543/32252100608", "-8318951675271/8063025152" ], "rational_weierstrass_points": 1, "minimal_model": { "f": [ "4", "10", "-4", "-1", "5", "-3", "1" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "129008402432", "minimal_discriminant_factored": "2^23 * 7 * 13^3", "conductor": "4732", "conductor_factored": "2^2 * 7 * 13^2", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "4732.c.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B4%2C10%2C-4%2C-1%2C5%2C-3%2C1%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "4732.c.1" }, "torsion": { "invariants": [ 28 ], "order": 28, "generators": [ { "order": 28, "mumford": [ "x^2 - 2/3*x + 8/3", "-23/9*x + 68/9", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 28, "point_count_primes": [ 3, 5, 11, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 28 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "cover", "over": "Q", "detail": "a nonconstant map of degree 3 from y^2 = g to the genus-1 curve v^2 = (1)u^3 + (0)u^2 + (1122741)u + (310814982) over Q (the elliptic curve 182a1), (x,y) -> (p/q, y*hh/q^2) with p = 891*x^3 - 891*x^2 + 2853*x + 891, q = x^3 - x^2 - x + 1, hh = 22464*x^3 - 11232*x^2 - 11232; degree-3 map to the elliptic curve 182a1 over Q [Genus2Torsion/verify_split_certificates.m, cover 9]" }, "isogenous_product": null, "cover_submitted": true, "cover_accepted": true, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Platonov–Zhgun–Petrunin 2013; Howe 2015", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 23 (source: \\cite{PZP2013,Howe2015})", "sources": [ "PZP2013", "Howe2015" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "4", "10", "-4", "-1", "5", "-3", "1" ], "h": [ "0", "1", "1" ], "claimed_group": [ 28 ], "claimed_class": "split", "cover": { "field": "", "cremona": "182a1", "cubic": [], "p": [ "891", "2853", "-891", "891" ], "q": [ "1", "-1", "-1", "1" ], "hh": [ "-11232", "0", "-11232", "22464" ], "note": "degree-3 map to the elliptic curve 182a1 over Q [Genus2Torsion/verify_split_certificates.m, cover 9]" }, "lmfdb_label_given": "4732.c.1" }, "source": { "kind": "import", "name": "paper-t2-023", "file": "table2.txt", "line": 29, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:37:19 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.12000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/28.b.log", "conductor_log": "data/logs/28.b.conductor.log", "conductor_cputime_seconds": 0.37 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "29.a", "group": [ 29 ], "group_key": "29", "group_bracket": "[29]", "order": 29, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "0", "-1", "2", "0", "-2", "1" ], "h": [ "1", "1" ], "f_str": "x^6 - 2*x^5 + 2*x^3 - x^2", "h_str": "x + 1", "g": [ "1", "2", "-3", "8", "0", "-8", "4" ], "g_str": "4*x^6 - 8*x^5 + 8*x^3 - 3*x^2 + 2*x + 1", "g_scale": "1", "discriminant": "4093640704", "discriminant_digits": 10, "igusa_clebsch": [ "-1216", "64768", "-35181568", "4093640704" ], "g2_invariants": [ "-4952198/61", "-61731/61", "551969/976" ], "rational_weierstrass_points": 1, "minimal_model": { "f": [ "1", "2", "0", "-2", "-1", "-1" ], "h": [ "0", "0", "1", "1" ] }, "minimal_discriminant": "999424", "minimal_discriminant_factored": "2^14 * 61", "conductor": "976", "conductor_factored": "2^4 * 61", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "976.a.999424.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/976/a/999424/1", "snapshot_label": "" }, "torsion": { "invariants": [ 29 ], "order": 29, "generators": [ { "order": 29, "mumford": [ "x - 1", "2*x^3", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 29, "point_count_primes": [ 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 29 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 3, "chi": "T^4 + 3*T^3 + 7*T^2 + 9*T + 9", "primes_tried_below": 500, "admissible_primes": 1, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 + 3*T^3 + 7*T^2 + 9*T + 9" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 44, "reducible_chi": 3 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Leprévost 1993", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 29 (source: \\cite{Leprevost1993})", "sources": [ "Leprevost1993" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "0", "-1", "2", "0", "-2", "1" ], "h": [ "1", "1" ], "claimed_group": [ 29 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "976.a.999424.1" }, "source": { "kind": "import", "name": "paper-t1-029", "file": "table1.txt", "line": 35, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:34 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.280000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/29.a.log", "conductor_log": "data/logs/29.a.conductor.log", "conductor_cputime_seconds": 0.21 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "30.a", "group": [ 30 ], "group_key": "30", "group_bracket": "[30]", "order": 30, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "-1", "16", "-12", "-10", "4", "2" ], "h": [ "0", "0", "1", "1" ], "f_str": "2*x^6 + 4*x^5 - 10*x^4 - 12*x^3 + 16*x^2 - x", "h_str": "x^3 + x^2", "g": [ "0", "-4", "64", "-48", "-39", "18", "9" ], "g_str": "9*x^6 + 18*x^5 - 39*x^4 - 48*x^3 + 64*x^2 - 4*x", "g_scale": "1", "discriminant": "-4522834132992", "discriminant_digits": 13, "igusa_clebsch": [ "50880", "149870016", "1930782605760", "-4522834132992" ], "g2_invariants": [ "-669112788800000/71001", "-228377318000/7889", "-1282659625/23667" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "2", "-5", "-10", "12", "16", "1" ], "h": [ "1", "1" ] }, "minimal_discriminant": "-1104207552", "minimal_discriminant_factored": "-1 * 2^6 * 3^7 * 7^3 * 23", "conductor": "8694", "conductor_factored": "2 * 3^3 * 7 * 23", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "8694.e.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C-1%2C16%2C-12%2C-10%2C4%2C2%5D%2C%5B0%2C0%2C1%2C1%5D%5D", "snapshot_label": "8694.e.1" }, "torsion": { "invariants": [ 30 ], "order": 30, "generators": [ { "order": 30, "mumford": [ "x^2 + 13/6*x - 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"conductor_magma": "7436" }, "lmfdb": { "label": "7436.a.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B160%2C80%2C166%2C182%2C116%2C30%2C10%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "7436.a.2" }, "torsion": { "invariants": [ 35 ], "order": 35, "generators": [ { "order": 35, "mumford": [ "x^2 + 6*x - 4", "-225*x + 100", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 35, "point_count_primes": [ 3, 7, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 35 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 91, "reducible_chi": 91 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution 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3^11 * 5^4 * 41", "conductor": "59040", "conductor_factored": "2^5 * 3^2 * 5 * 41", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "59040.r.3", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B9%2C6%2C-14%2C13%2C30%2C3%5D%2C%5B0%2C0%2C1%5D%5D", "snapshot_label": "59040.r.3" }, "torsion": { "invariants": [ 36 ], "order": 36, "generators": [ { "order": 36, "mumford": [ "x^2 - 14*x - 12", "-244*x - 198", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 36, "point_count_primes": [ 7, 11, 13, 17, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 36 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 11, "chi": "T^4 - 2*T^3 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"https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:40 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.360000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/39.a.log", "conductor_log": "data/logs/39.a.conductor.log", "conductor_cputime_seconds": 0.28 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "40.a", "group": [ 40 ], "group_key": "40", "group_bracket": "[40]", "order": 40, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "0", "-2", "-2", "4", "3" ], "h": [ "1", "1" ], "f_str": "3*x^5 + 4*x^4 - 2*x^3 - 2*x^2", "h_str": "x + 1", "g": 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"generators_note": "", "generators_submitted": 0, "point_count_gcd": 40, "point_count_primes": [ 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 40 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 17, "chi": "T^4 + T^3 + 12*T^2 + 17*T + 289", "primes_tried_below": 500, "admissible_primes": 5, "reducible_chi": 3 }, "q_simple": { "certified": true, "prime": 7, "chi": "T^4 - 10*T^2 + 49" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 43, "reducible_chi": 15 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Elkies 2002", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 37 (source: \\cite{Elkies2002})", 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"schema": "genus-2-torsion/curve/1", "id": "40.b", "group": [ 40 ], "group_key": "40", "group_bracket": "[40]", "order": 40, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "66", "0", "12", "0", "2" ], "h": [ "0", "1", "0", "1" ], "f_str": "2*x^4 + 12*x^2 + 66", "h_str": "x^3 + x", "g": [ "264", "0", "49", "0", "10", "0", "1" ], "g_str": "x^6 + 10*x^4 + 49*x^2 + 264", "g_scale": "1", "discriminant": "-11916368249094144", "discriminant_digits": 17, "igusa_clebsch": [ "-71200", "66989632", "-1515605756416", "-11916368249094144" ], "g2_invariants": [ "54531830556640625/2841083586", "597251939828125/947027862", "283532195000/14348907" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "66", "0", "12", "0", "2" ], "h": [ "0", "1", "0", "1" ] }, "minimal_discriminant": "-2909269592064", "minimal_discriminant_factored": "-1 * 2^11 * 3^17 * 11", "conductor": "3168", "conductor_factored": "66 * 48 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 66 * 48 for the elliptic curves ['1', '0', '0', '115', '561'] and ['0', '1', '0', '16', '180'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "3168" }, "lmfdb": { "label": "3168.i.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B66%2C0%2C12%2C0%2C2%5D%2C%5B0%2C1%2C0%2C1%5D%5D", "snapshot_label": "3168.i.1" }, "torsion": { "invariants": [ 40 ], "order": 40, "generators": [ { "order": 40, "mumford": [ "x^2 - 9/2*x + 25/2", "-45/4*x + 207/4", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 80, "point_count_primes": [ 5, 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 40 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x : y : -z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "0", "0", "115", "561" ], [ "0", "1", "0", "16", "180" ] ], "conductors": [ "66", "48" ], "conductor": "3168", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { 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Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.840000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/40.b.log", "conductor_log": "data/logs/40.b.conductor.log", "conductor_cputime_seconds": 0.14 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "45.a", "group": [ 45 ], "group_key": "45", "group_bracket": "[45]", "order": 45, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "168300000000", "0", "49996210000", "0", "29240200", "0", "13981" ], "h": [ "0" ], "f_str": "13981*x^6 + 29240200*x^4 + 49996210000*x^2 + 168300000000", "h_str": "0", "g": [ "673200000000", "0", "199984840000", "0", "116960800", "0", "55924" ], "g_str": "55924*x^6 + 116960800*x^4 + 199984840000*x^2 + 673200000000", "g_scale": "1", "discriminant": 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"conductor": "602835300", "conductor_factored": "15810 * 38130 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 15810 * 38130 for the elliptic curves ['1', '0', '0', '-35358150', '80924422500'] and ['1', '0', '0', '176730', '-43451388'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "602835300" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B168300000000%2C0%2C49996210000%2C0%2C29240200%2C0%2C13981%5D%2C%5B0%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 45 ], "order": 45, "generators": [ { "order": 45, "mumford": [ "x^2 - 440/19*x + 24100/19", "58709200/361*x - 38236000/361", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 45, "point_count_primes": [ 7, 13, 19, 23, 29, 37, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131 ], "claimed": [ 45 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 88, "reducible_chi": 88 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x : y : -z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "0", "0", "-35358150", "80924422500" ], [ "1", "0", "0", "176730", "-43451388" ] ], "conductors": [ "15810", "38130" ], "conductor": "602835300", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 39, "reducible_chi": 39 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Balakrishnan–Najman–Shnidman–Sutherland 2026", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 28 (source: \\cite{HLP2000})", "sources": [ "HLP2000" ], "rm": false, "notes": "lane8 anchor (HLP sec 3.6 family)", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "168300000000", "0", "49996210000", "0", "29240200", "0", "13981" ], "h": [ "0" ], "claimed_group": [ 45 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "import", "name": "paper-t2-028", "file": "table2.txt", "line": 34, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { 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"discriminant_digits": 13, "igusa_clebsch": [ "5408", "5290240", "4401143552", "-5119792054272" ], "g2_invariants": [ "-8822943478336/78121827", "232111591192/26040609", "-10859577664/8680203" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "3", "0", "9", "0", "-4" ], "h": [ "0", "1", "0", "1" ] }, "minimal_discriminant": "-1249949232", "minimal_discriminant_factored": "-1 * 2^4 * 3^13 * 7^2", "conductor": "1764", "conductor_factored": "21 * 84 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 21 * 84 for the elliptic curves ['1', '0', '0', '-39', '90'] and ['0', '1', '0', '-28', '-28'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "1764" }, "lmfdb": { "label": "1764.a.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B3%2C0%2C9%2C0%2C-4%5D%2C%5B0%2C1%2C0%2C1%5D%5D", "snapshot_label": "1764.a.1" }, "torsion": { "invariants": [ 48 ], "order": 48, "generators": [ { "order": 48, "mumford": [ "x^2 - 3/2*x - 11/2", "-3/4*x + 15/4", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 48, "point_count_primes": [ 5, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 48 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x : y : -z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "0", "0", "-39", "90" ], [ "0", "1", "0", "-28", "-28" ] ], "conductors": [ "21", "84" ], "conductor": "1764", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Platonov–Petrunin 2012b", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 29 (source: \\cite{Howe2015, PP2012b,Platonov2014})", "sources": [ "Howe2015", "PP2012b", "Platonov2014" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "3", "0", "9", "0", "-4" ], "h": [ "0", "1", "0", "1" ], "claimed_group": [ 48 ], 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"accepted_because": "new group", "curve": { "f": [ "1", "-7", "25", "-24", "25", "-7", "1" ], "h": [ "0", "1", "1" ], "f_str": "x^6 - 7*x^5 + 25*x^4 - 24*x^3 + 25*x^2 - 7*x + 1", "h_str": "x^2 + x", "g": [ "4", "-28", "101", "-94", "101", "-28", "4" ], "g_str": "4*x^6 - 28*x^5 + 101*x^4 - 94*x^3 + 101*x^2 - 28*x + 4", "g_scale": "1", "discriminant": "-105672301608960000", "discriminant_digits": 18, "igusa_clebsch": [ "-82680", "164134980", "-4357254195000", "-105672301608960000" ], "g2_invariants": [ "776365295912245/169869312", "3320217188119/28311552", "-657488585/1572864" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "1", "-7", "25", "-24", "25", "-7", "1" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "-25798901760000", "minimal_discriminant_factored": "-1 * 2^21 * 3^9 * 5^4", "conductor": "13500", "conductor_factored": "150 * 90 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 150 * 90 for the elliptic curves ['1', '0', '0', '-28', '272'] and ['1', '-1', '1', '-122', '1721'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "13500" }, "lmfdb": { "label": "13500.r.4", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B1%2C-7%2C25%2C-24%2C25%2C-7%2C1%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "13500.r.4" }, "torsion": { "invariants": [ 60 ], "order": 60, "generators": [ { "order": 60, "mumford": [ "x - 2", "-2*x^3 + 40", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 120, "point_count_primes": [ 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 60 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (z : y : x) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "0", "0", "-28", "272" ], [ "1", "-1", "1", "-122", "1721" ] ], "conductors": [ "150", "90" ], "conductor": "13500", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 30 (source: \\cite{HLP2000})", "sources": [ "HLP2000" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "1", "-7", "25", "-24", "25", "-7", "1" ], "h": [ "0", "1", "1" ], "claimed_group": [ 60 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "13500.r.4" }, "source": { "kind": "import", "name": "paper-t2-030", "file": "table2.txt", "line": 36, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:37:29 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.12000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/60.a.log", "conductor_log": "data/logs/60.a.conductor.log", "conductor_cputime_seconds": 0.51 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "63.a", "group": [ 63 ], "group_key": "63", "group_bracket": "[63]", "order": 63, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "-146398496", "0", "79136353", "0", "-197570", "0", "897" ], "h": [ "0" ], "f_str": "897*x^6 - 197570*x^4 + 79136353*x^2 - 146398496", "h_str": "0", "g": [ "-585593984", "0", "316545412", "0", "-790280", "0", "3588" ], "g_str": "3588*x^6 - 790280*x^4 + 316545412*x^2 - 585593984", "g_scale": "1", "discriminant": "19875447372006204730833007077824981972668459047020905769288447797009317888", "discriminant_digits": 74, "igusa_clebsch": [ "4506818822627840", "5469064735224145458735953231872", "4887936592007094287626475631766143452240150528", "19875447372006204730833007077824981972668459047020905769288447797009317888" ], "g2_invariants": [ "1275622973999186561187700559463173332023112802190625/109088440895853914785379999772145736865470742528", "-7415334447747926554091695150879908654480758875/4600558404852138781434716589581044908294144", "1021590225322496229867630123004633925/18532634573667637841873990213173248" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "-1054044", "4921494", "4872774", "-96543", "-46029", "2691", "897" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "4519153520051578364207900239722444433179693771085318478364672", "minimal_discriminant_factored": "2^41 * 3^15 * 13^3 * 19^5 * 23^22 * 29", "conductor": "63693396", "conductor_factored": "42978 * 1482 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 42978 * 1482 for the elliptic curves ['1', '0', '0', '-459769306', '3796241996132'] and ['1', '0', '0', '12948', '421776'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "63693396" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-146398496%2C0%2C79136353%2C0%2C-197570%2C0%2C897%5D%2C%5B0%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 63 ], "order": 63, "generators": [ { "order": 63, "mumford": [ "x^2 - 368/37*x + 475/37", "14938960/1369*x - 31037488/1369", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 63, "point_count_primes": [ 5, 7, 11, 17, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127 ], "claimed": [ 63 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 89, "reducible_chi": 89 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x : y : -z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "0", "0", "-459769306", "3796241996132" ], [ "1", "0", "0", "12948", "421776" ] ], "conductors": [ "42978", "1482" ], "conductor": "63693396", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 40, "reducible_chi": 40 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Howe–Leprévost–Poonen 2000", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 31 (source: \\cite{HLP2000})", "sources": [ "HLP2000" ], "rm": false, "notes": "HLP eq (4) curve / lane8", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-146398496", "0", "79136353", "0", "-197570", "0", "897" ], "h": [ "0" ], "claimed_group": [ 63 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "import", "name": "paper-t2-031", "file": "table2.txt", "line": 37, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:37:32 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.15000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/63.a.log", "conductor_log": "data/logs/63.a.conductor.log", "conductor_cputime_seconds": 1.54 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "70.a", "group": [ 70 ], "group_key": "70", "group_bracket": "[70]", "order": 70, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "179", "425", "-51", "1" ], "h": [ "110", "-41", "-3", "2" ], "f_str": "x^3 - 51*x^2 + 425*x + 179", "h_str": "2*x^3 - 3*x^2 - 41*x + 110", "g": [ "12816", "-7320", "817", "690", "-155", "-12", "4" ], "g_str": "4*x^6 - 12*x^5 - 155*x^4 + 690*x^3 + 817*x^2 - 7320*x + 12816", "g_scale": "1", "discriminant": "1200389416240068618834935808", "discriminant_digits": 28, "igusa_clebsch": [ "-3907000", "3746123344516", "-4310616529541439352", "1200389416240068618834935808" ], "g2_invariants": [ "-27782231202000009368896484375/293063822324235502645248", "17110083929150941232421875/1480120314768866174976", "11683032545125261515625/7475355125095283712" ], "rational_weierstrass_points": 1, "minimal_model": { "f": [ "1089", "-231", "309", "-98", "-9", "9", "1" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "293063822324235502645248", "minimal_discriminant_factored": "2^12 * 3^24 * 11^7 * 13", "conductor": "56628", "conductor_factored": "2^2 * 3^2 * 11^2 * 13", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B179%2C425%2C-51%2C1%5D%2C%5B110%2C-41%2C-3%2C2%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 70 ], "order": 70, "generators": [ { "order": 70, "mumford": [ "x^2 - 10*x - 12", "177*x + 48", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 70, "point_count_primes": [ 5, 7, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 70 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 91, "reducible_chi": 91 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "cover", "over": "Q", "detail": "a nonconstant map of degree 3 from y^2 = g to the genus-1 curve v^2 = (1)u^3 + (0)u^2 + (-7483623723)u + (249446508217254) over Q (the elliptic curve 858k1), (x,y) -> (p/q, y*hh/q^2) with p = -15693*x^3 + 78465*x^2 - 5487933/4*x + 4954371, q = x^3 - 5*x^2 - 55/4*x + 89, hh = 9526572*x^3 - 80975862*x^2 + 190531440*x - 161951724; degree-3 map to the elliptic curve 858k1 over Q [Genus2Torsion/verify_split_certificates.m, cover 17]" }, "isogenous_product": null, "cover_submitted": true, "cover_accepted": true, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 42, "reducible_chi": 42 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Howe 2015", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 32 (source: \\cite{Howe2015})", "sources": [ "Howe2015" ], "rm": false, "notes": "Howe 2015 Thm 2.4", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "179", "425", "-51", "1" ], "h": [ "110", "-41", "-3", "2" ], "claimed_group": [ 70 ], "claimed_class": "split", "cover": { "field": "", "cremona": "858k1", "cubic": [], "p": [ "4954371", "-5487933/4", "78465", "-15693" ], "q": [ "89", "-55/4", "-5", "1" ], "hh": [ "-161951724", "190531440", "-80975862", "9526572" ], "note": "degree-3 map to the elliptic curve 858k1 over Q [Genus2Torsion/verify_split_certificates.m, cover 17]" }, "lmfdb_label_given": "" }, "source": { "kind": "import", "name": "paper-t2-032", "file": "table2.txt", "line": 38, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:37:33 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.910000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/70.a.log", "conductor_log": "data/logs/70.a.conductor.log", "conductor_cputime_seconds": 0.24 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.a", "group": [ 2, 2 ], "group_key": "2.2", "group_bracket": "[2,2]", "order": 4, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "1", "16", "72", "33", "4" ], "h": [ "0", "1" ], "f_str": "4*x^5 + 33*x^4 + 72*x^3 + 16*x^2 + x", "h_str": "x", "g": [ "0", "4", "65", "288", "132", "16" ], "g_str": "16*x^5 + 132*x^4 + 288*x^3 + 65*x^2 + 4*x", "g_scale": "1", "discriminant": "-475136", "discriminant_digits": 6, "igusa_clebsch": [ "362944", "12942400", "1542241461760", "-121634816" ], "g2_invariants": [ "-187693059992988715232/29", "-7808250185554819143/29", "-432507850151022641/29" ], "rational_weierstrass_points": 3, "minimal_model": { "f": [ "0", "-4", "33", "-72", "16", "-1" ], "h": [ "0", "0", "1" ] }, "minimal_discriminant": "-29696", "minimal_discriminant_factored": "-1 * 2^10 * 29", "conductor": "464", "conductor_factored": "2^4 * 29", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "464.a.29696.2", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/464/a/29696/2", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 2 ], "order": 4, "generators": [ { "order": 2, "mumford": [ "x^2 + 4*x", "0", "2" ] }, { "order": 2, "mumford": [ "x", "0", "1" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 16, "point_count_primes": [ 3, 5, 7, 11, 13, 17, 19, 23, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 2, 2 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 3, "chi": "T^4 + T^3 + 2*T^2 + 3*T + 9", "primes_tried_below": 500, "admissible_primes": 1, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 + T^3 + 2*T^2 + 3*T + 9" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 44, "reducible_chi": 14 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 38 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "1", "16", "72", "33", "4" ], "h": [ "0", "1" ], "claimed_group": [ 2, 2 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "464.a.29696.2" }, "source": { "kind": "import", "name": "paper-t1-038", "file": "table1.txt", "line": 44, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:41 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.300000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.a.log", "conductor_log": "data/logs/2.2.a.conductor.log", "conductor_cputime_seconds": 0.25 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.b", "group": [ 2, 2 ], "group_key": "2.2", "group_bracket": "[2,2]", "order": 4, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-60", "-1", "59", "-1", "59", "0", "-60" ], "h": [ "1", "1", "1", "1" ], "f_str": "-60*x^6 + 59*x^4 - x^3 + 59*x^2 - x - 60", "h_str": "x^3 + x^2 + x + 1", "g": [ "-239", "-2", "239", "0", "239", "2", "-239" ], "g_str": "-239*x^6 + 2*x^5 + 239*x^4 + 239*x^2 - 2*x - 239", "g_scale": "1", "discriminant": "-48859625810781274112", "discriminant_digits": 20, "igusa_clebsch": [ "-14623136", "3341101424128", "-18321731531444224000", "-48859625810781274112" ], "g2_invariants": [ "39854989545308934871316672186/23298085122481", "4981917301119231624074738903/93192340489924", "-47714060676026481/186384680979848" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "-60", "-1", "59", "-1", "59", "0", "-60" ], "h": [ "1", "1", "1", "1" ] }, "minimal_discriminant": "-11928619582710272", "minimal_discriminant_factored": "-1 * 2^9 * 13^12", "conductor": "256", "conductor_factored": "2^8", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "256.a.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-60%2C-1%2C59%2C-1%2C59%2C0%2C-60%5D%2C%5B1%2C1%2C1%2C1%5D%5D", "snapshot_label": "256.a.2" }, "torsion": { "invariants": [ 2, 2 ], "order": 4, "generators": [ { "order": 2, "mumford": [ "x^2 - 1", "0", "2" ] }, { "order": 2, "mumford": [ "x^2 + 1", "0", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 20, "point_count_primes": [ 3, 5, 7, 11, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 2, 2 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 31 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 + 2*T^3 + 2*T^2 + 6*T + 9" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "geometric_involution", "over": "Qbar", "detail": "the geometric automorphism group has order 8 (5 involutions), so the curve is bielliptic over Qbar" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 44, "reducible_chi": 15 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 33 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-60", "-1", "59", "-1", "59", "0", "-60" ], "h": [ "1", "1", "1", "1" ], "claimed_group": [ 2, 2 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "256.a.2" }, "source": { "kind": "import", "name": "paper-t2-033", "file": "table2.txt", "line": 39, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:37:34 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.860000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.b.log", "conductor_log": "data/logs/2.2.b.conductor.log", "conductor_cputime_seconds": 0.13 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.c", "group": [ 2, 2 ], "group_key": "2.2", "group_bracket": "[2,2]", "order": 4, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "0", "3", "28", "72", "28", "3" ], "h": [ "0", "1", "1" ], "f_str": "3*x^5 + 28*x^4 + 72*x^3 + 28*x^2 + 3*x", "h_str": "x^2 + x", "g": [ "0", "12", "113", "290", "113", "12" ], "g_str": "12*x^5 + 113*x^4 + 290*x^3 + 113*x^2 + 12*x", "g_scale": "1", "discriminant": "6842880", "discriminant_digits": 7, "igusa_clebsch": [ "306056", "27393028", "2746933168904", "985374720" ], "g2_invariants": [ "81951056110393451083057/240570", "188813894774599018858/13365", "7001861848004294/9" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "0", "3", "28", "72", "28", "3" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "240570", "minimal_discriminant_factored": "2 * 3^7 * 5 * 11", "conductor": "990", "conductor_factored": "15 * 66 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 15 * 66 for the elliptic curves ['1', '1', '1', '-135', '-660'] and ['1', '1', '1', '-22', '-49'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "990" }, "lmfdb": { "label": "990.a.240570.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/990/a/240570/1", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 2 ], "order": 4, "generators": [ { "order": 2, "mumford": [ "x", "0", "1" ] }, { "order": 2, "mumford": [ "x^2 + 19/4*x + 1", "0", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "the submitted generators generate a subgroup of order 4, but the point-count bound is 32; TorsionSubgroup was run", "generators_submitted": 2, "point_count_gcd": 32, "point_count_primes": [ 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 2, 2 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 91, "reducible_chi": 91 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (z : y : x) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "1", "1", "-135", "-660" ], [ "1", "1", "1", "-22", "-49" ] ], "conductors": [ "15", "66" ], "conductor": "990", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 42, "reducible_chi": 42 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "LMFDB 990.a.240570.1, https://www.lmfdb.org/Genus2Curve/Q/990/a/240570/1", "sources": [], "rm": false, "notes": "Q-split witness: the census has only geometrically simple and Q-simple (geometrically split) curves with this torsion group. Not a new curve. LMFDB: torsion [2,2], not simple over Q (is_simple_base = false). Split over Q by the involution (x, y) -> (1/x, y/x^3). Output of pipeline/magma/prepare_submission.m (Magma V2.29-4): J(Q)_tors = [2,2]; gcd of #J(F_p) over 25 good primes = 32 (not sharp); no strict prime below 500; conductor 990 = 2 * 3^2 * 5 * 11.", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "0", "3", "28", "72", "28", "3" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 2 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "issue", "number": 5, "url": "https://github.com/Genus-2-torsion/Genus-2-torsion.github.io/issues/5" }, "dates": { "submitted": "2026-10-14", "verified": "2026-10-14 10:49:12 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.43000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.c.log", "conductor_log": "data/logs/2.2.c.conductor.log", "conductor_cputime_seconds": 0.35 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.4.a", "group": [ 2, 4 ], "group_key": "2.4", "group_bracket": "[2,4]", "order": 8, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "-1", "0", "16", "-8", "1" ], "h": [ "0", "1" ], "f_str": "x^5 - 8*x^4 + 16*x^3 - x", "h_str": "x", "g": [ "0", "-4", "1", "64", "-32", "4" ], "g_str": "4*x^5 - 32*x^4 + 64*x^3 + x^2 - 4*x", "g_scale": "1", "discriminant": "255232", "discriminant_digits": 6, "igusa_clebsch": [ "24448", "769024", "6279257536", "4083712" ], "g2_invariants": [ "266542673508171776/997", "10877317101649920/997", "578694117523712/997" ], "rational_weierstrass_points": 3, "minimal_model": { "f": [ "0", "-1", "0", "16", "-8", "1" ], "h": [ "0", "1" ] }, "minimal_discriminant": "997", "minimal_discriminant_factored": "997", "conductor": "997", "conductor_factored": "997", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "997.a.997.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/997/a/997/1", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 4 ], "order": 8, "generators": [ { "order": 2, "mumford": [ "x", "0", "1" ] }, { "order": 4, "mumford": [ "x^2 - 4*x + 1", "x", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 16, "point_count_primes": [ 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101 ], "claimed": [ 2, 4 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 3, "chi": "T^4 + T^3 + 2*T^2 + 3*T + 9", "primes_tried_below": 500, "admissible_primes": 1, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 + T^3 + 2*T^2 + 3*T + 9" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 45, "reducible_chi": 12 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 39 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "-1", "0", "16", "-8", "1" ], "h": [ "0", "1" ], "claimed_group": [ 2, 4 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "997.a.997.1" }, "source": { "kind": "import", "name": "paper-t1-039", "file": "table1.txt", "line": 45, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:42 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.310000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.4.a.log", "conductor_log": "data/logs/2.4.a.conductor.log", "conductor_cputime_seconds": 0.12 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.4.b", "group": [ 2, 4 ], "group_key": "2.4", "group_bracket": "[2,4]", "order": 8, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-16", "6", "-6", "11", "1", "3" ], "h": [ "1", "0", "1" ], "f_str": "3*x^5 + x^4 + 11*x^3 - 6*x^2 + 6*x - 16", "h_str": "x^2 + 1", "g": [ "-63", "24", "-22", "44", "5", "12" ], "g_str": "12*x^5 + 5*x^4 + 44*x^3 - 22*x^2 + 24*x - 63", "g_scale": "1", "discriminant": "6370099200", "discriminant_digits": 10, "igusa_clebsch": [ "24896", "99409984", "1157149366784", "917294284800" ], "g2_invariants": [ "71258716167592/54675", "-2066769304459/24300", "-1625338861/36" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "-16", "6", "-6", "11", "1", "3" ], "h": [ "1", "0", "1" ] }, "minimal_discriminant": "223948800", "minimal_discriminant_factored": "2^12 * 3^7 * 5^2", "conductor": "225", "conductor_factored": "15 * 15 (product of the elliptic conductors)", "conductor_note": "Magma's genus-2 Conductor gave 230400 (Ogg's formula at 2, no correctness guarantee); the rigorous value is N(E1) N(E2) = 15 * 15 for the elliptic curves ['1', '1', '1', '-5', '2'] and ['1', '1', '1', '-135', '-660'] to whose product J is isogenous over Q", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "230400" }, "lmfdb": { "label": "225.a.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-16%2C6%2C-6%2C11%2C1%2C3%5D%2C%5B1%2C0%2C1%5D%5D", "snapshot_label": "225.a.1" }, "torsion": { "invariants": [ 2, 4 ], "order": 8, "generators": [ { "order": 2, "mumford": [ "x - 1", "0", "1" ] }, { "order": 4, "mumford": [ "x^2 + x + 2", "x + 3", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 64, "point_count_primes": [ 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 2, 4 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x + 3*z : -x^3 + 3*x^2*z - 3*x*z^2 + 8*y + 9*z^3 : x - z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "1", "1", "-5", "2" ], [ "1", "1", "1", "-135", "-660" ] ], "conductors": [ "15", "15" ], "conductor": "225", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 34 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-16", "6", "-6", "11", "1", "3" ], "h": [ "1", "0", "1" ], "claimed_group": [ 2, 4 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "225.a.1" }, "source": { "kind": "import", "name": "paper-t2-034", "file": "table2.txt", "line": 40, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:37:36 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.01000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.4.b.log", "conductor_log": "data/logs/2.4.b.conductor.log", "conductor_cputime_seconds": 0.21 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "2.4.c", "group": [ 2, 4 ], "group_key": "2.4", "group_bracket": "[2,4]", "order": 8, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "0", "-1", "11", "35", "-123", "-171", "405" ], "h": [ "0" ], "f_str": "405*x^6 - 171*x^5 - 123*x^4 + 35*x^3 + 11*x^2 - x", "h_str": "0", "g": [ "0", "-4", "44", "140", "-492", "-684", "1620" ], "g_str": "1620*x^6 - 684*x^5 - 492*x^4 + 140*x^3 + 44*x^2 - 4*x", "g_scale": "1", "discriminant": "1107783872227049472", "discriminant_digits": 19, "igusa_clebsch": [ "573408", "16803800064", "2488000326598656", "1107783872227049472" ], "g2_invariants": [ "31286450905550951/4472832", "51328501722964015/966131712", "17820045740415853/52171112448" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "405", "-171", "-123", "35", "11", "-1" ], "h": [] }, "minimal_discriminant": "270455046930432", "minimal_discriminant_factored": "2^24 * 3^11 * 7 * 13", "conductor": "2457", "conductor_factored": "3^3 * 7 * 13", "conductor_note": "taken from the LMFDB label 2457.b.3 (Magma's genus-2 Conductor failed on this curve: Fibre blowups over number fields not yet implemented)", "conductor_source": "lmfdb label" }, "lmfdb": { "label": "2457.b.3", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C-1%2C11%2C35%2C-123%2C-171%2C405%5D%2C%5B0%5D%5D", "snapshot_label": "2457.b.3" }, "torsion": { "invariants": [ 2, 4 ], "order": 8, "generators": [ { "order": 2, "mumford": [ "x^2 + 1/3*x", "0", "2" ] }, { "order": 4, "mumford": [ "x^2 - 1/33*x - 1/33", "-348/121*x + 4/121", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "the submitted generators generate a subgroup of order 8, but the point-count bound is 16; TorsionSubgroup was run", "generators_submitted": 2, "point_count_gcd": 16, "point_count_primes": [ 11, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113 ], "claimed": [ 2, 4 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 90, "reducible_chi": 49 }, "q_simple": { "certified": true, "prime": 11, "chi": "T^4 - 10*T^2 + 121" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "richelot", "over": "Q", "detail": "a chain of two (2,2)-isogenies over Q reaches a Weil restriction of an elliptic curve E over Q(sqrt(-3)) (the surface is E x E^sigma)" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 41, "reducible_chi": 24 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": 2026, "historical": false }, "reference": "LMFDB extended genus 2 database (Booker–Sutherland), curve 2457.b.3 (table g2c_curves_new, label 2457.b3 on devmirror.lmfdb.xyz, read 2026-10-16)", "sources": [ "BookerSutherland" ], "rm": false, "notes": "First Q-simple, geometrically split example of [2,4] for the census (the gsplit cell is empty). Found by querying the LMFDB genus 2 data (devmirror.lmfdb.xyz, g2c_curves_new: is_simple_base = true, is_simple_geom = false, torsion_subgroup = [2,4]; geom_end_alg = Q x Q, end_alg = Q; smallest conductor among the curves of this group that the census verifier certifies). LMFDB conductor 2457. Checked locally with the census's own pipeline/magma/verify_lib.m (Magma V2.30-1): J(Q)_tors = [2,4], class gsplit, Q-simplicity prime 11 (chi = T^4 - 10*T^2 + 121), splitness certificate: richelot over Q (a chain of two (2,2)-isogenies over Q reaches a Weil restriction of an elliptic curve E over Q(sqrt(-3)) (the surface is E x E^sigma)). Submitted by F. Najman; the curve is due to the database cited under Reference. Rigorous conductor: J is (2,2)-isogenous over Q to Res_(K/Q) E with K the quadratic field of discriminant -3, so N(J) = |d_K|^2 * Norm(f_E) = 9 * 273 = 2457 (computed with Magma; conductor is an isogeny invariant).", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "0", "-1", "11", "35", "-123", "-171", "405" ], "h": [ "0" ], "claimed_group": [ 2, 4 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "2457.b.3" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_2.4.json" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-17 06:12:08 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "1.43000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.4.c.log", "conductor_log": "data/logs/2.4.c.conductor.log", "conductor_cputime_seconds": null }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.6.a", "group": [ 2, 6 ], "group_key": "2.6", "group_bracket": "[2,6]", "order": 12, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "0", "-2", "-1", "4", "4" ], "h": [ "1" ], "f_str": "4*x^5 + 4*x^4 - x^3 - 2*x^2", "h_str": "1", "g": [ "1", "0", "-8", "-4", "16", "16" ], "g_str": "16*x^5 + 16*x^4 - 4*x^3 - 8*x^2 + 1", "g_scale": "1", "discriminant": "-720896", "discriminant_digits": 6, "igusa_clebsch": [ "2144", "-118784", "-62783488", "-184549376" ], "g2_invariants": [ "-1350125107/44", "-636113745/352", "-68955529/704" ], "rational_weierstrass_points": 3, "minimal_model": { "f": [ "-1", "0", "-2", "-3", "1", "1" ], "h": [ "1", "1", "1", "1" ] }, "minimal_discriminant": "-45056", "minimal_discriminant_factored": "-1 * 2^12 * 11", "conductor": "704", "conductor_factored": "2^6 * 11", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "704.a.45056.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/704/a/45056/1", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 6 ], "order": 12, "generators": [ { "order": 2, "mumford": [ "x^2 - 1/4", "0", "2" ] }, { "order": 6, "mumford": [ "x^2 - 1/2*x", "2*x - 1", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 12, "point_count_primes": [ 3, 5, 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 2, 6 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 23, "chi": "T^4 - 3*T^3 - 2*T^2 - 69*T + 529", "primes_tried_below": 500, "admissible_primes": 7, "reducible_chi": 4 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 + T^3 - 2*T^2 + 3*T + 9" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 44, "reducible_chi": 17 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 40 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "0", "-2", "-1", "4", "4" ], "h": [ "1" ], "claimed_group": [ 2, 6 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "704.a.45056.1" }, "source": { "kind": "import", "name": "paper-t1-040", "file": "table1.txt", "line": 46, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:42 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.310000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.6.a.log", "conductor_log": "data/logs/2.6.a.conductor.log", "conductor_cputime_seconds": 0.18 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.6.b", "group": [ 2, 6 ], "group_key": "2.6", "group_bracket": "[2,6]", "order": 12, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-23", "13", "-14", "-3", "1", "-2" ], "h": [ "0", "1", "1" ], "f_str": "-2*x^5 + x^4 - 3*x^3 - 14*x^2 + 13*x - 23", "h_str": "x^2 + x", "g": [ "-92", "52", "-55", "-10", "5", "-8" ], "g_str": "-8*x^5 + 5*x^4 - 10*x^3 - 55*x^2 + 52*x - 92", "g_scale": "1", "discriminant": "6826389405696", "discriminant_digits": 13, "igusa_clebsch": [ "-11640", "85894020", "-92205692280", "436888921964544" ], "g2_invariants": [ "-26835438303125/438939648", "3833797020625/164602368", "-11983537625/2893401" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "-23", "13", "-14", "-3", "1", "-2" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "106662334464", "minimal_discriminant_factored": "2^12 * 3^12 * 7^2", "conductor": "196", "conductor_factored": "14 * 14 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 14 * 14 for the elliptic curves ['1', '0', '1', '-1', '0'] and ['1', '0', '1', '-171', '-874'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "196" }, "lmfdb": { "label": "196.a.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-23%2C13%2C-14%2C-3%2C1%2C-2%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "196.a.2" }, "torsion": { "invariants": [ 2, 6 ], "order": 12, "generators": [ { "order": 2, "mumford": [ "x^2 - 13/8*x + 23/8", "0", "2" ] }, { "order": 6, "mumford": [ "x^2 - 3/2*x + 2", "-5/2*x + 4", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 36, "point_count_primes": [ 5, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 2, 6 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x - 5/2*z : 1/8*x^3 + 3/4*x^2*z + 3/2*x*z^2 + 27/8*y + 35/8*z^3 : -1/2*x - z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "0", "1", "-1", "0" ], [ "1", "0", "1", "-171", "-874" ] ], "conductors": [ "14", "14" ], "conductor": "196", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 35 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-23", "13", "-14", "-3", "1", "-2" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 6 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "196.a.2" }, "source": { "kind": "import", "name": "paper-t2-035", "file": "table2.txt", "line": 41, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:37:37 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.19000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.6.b.log", "conductor_log": "data/logs/2.6.b.conductor.log", "conductor_cputime_seconds": 0.22 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "2.6.c", "group": [ 2, 6 ], "group_key": "2.6", "group_bracket": "[2,6]", "order": 12, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "0", "1", "-4", "-13", "-9", "-1" ], "h": [ "0" ], "f_str": "-x^5 - 9*x^4 - 13*x^3 - 4*x^2 + x", "h_str": "0", "g": [ "0", "4", "-16", "-52", "-36", "-4" ], "g_str": "-4*x^5 - 36*x^4 - 52*x^3 - 16*x^2 + 4*x", "g_scale": "1", "discriminant": "157351936", "discriminant_digits": 9, "igusa_clebsch": [ "6368", "2308096", "3735904256", "2517630976" ], "g2_invariants": [ "1248318403996/2401", "9291226221/4802", "-23245787/9604" ], "rational_weierstrass_points": 3, "minimal_model": { "f": [ "0", "1", "-4", "-13", "-9", "-1" ], "h": [] }, "minimal_discriminant": "614656", "minimal_discriminant_factored": "2^8 * 7^4", "conductor": "784", "conductor_factored": "2^4 * 7^2", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "784.b.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C1%2C-4%2C-13%2C-9%2C-1%5D%2C%5B0%5D%5D", "snapshot_label": "784.b.1" }, "torsion": { "invariants": [ 2, 6 ], "order": 12, "generators": [ { "order": 2, "mumford": [ "x^2 + x", "0", "2" ] }, { "order": 6, "mumford": [ "x^2 + 9*x - 1", "-11*x + 1", "2" ] } ], "method": "generators", "generators_note": "", "generators_submitted": 2, "point_count_gcd": 12, "point_count_primes": [ 3, 5, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 2, 6 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 93, "reducible_chi": 38 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 + T^3 - 2*T^2 + 3*T + 9" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "geometric_involution", "over": "Qbar", "detail": "the geometric automorphism group has order 12 (7 involutions), so the curve is bielliptic over Qbar" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 44, "reducible_chi": 18 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": 2026, "historical": false }, "reference": "LMFDB extended genus 2 database (Booker–Sutherland), curve 784.b.1 (table g2c_curves_new, label 784.b1 on devmirror.lmfdb.xyz, read 2026-10-16)", "sources": [ "BookerSutherland" ], "rm": false, "notes": "First Q-simple, geometrically split example of [2,6] for the census (the gsplit cell is empty). Found by querying the LMFDB genus 2 data (devmirror.lmfdb.xyz, g2c_curves_new: is_simple_base = true, is_simple_geom = false, torsion_subgroup = [2,6]; geom_end_alg = M_2(Q), end_alg = CM; smallest conductor among the curves of this group that the census verifier certifies). LMFDB conductor 784. Checked locally with the census's own pipeline/magma/verify_lib.m (Magma V2.30-1): J(Q)_tors = [2,6], class gsplit, Q-simplicity prime 3 (chi = T^4 + T^3 - 2*T^2 + 3*T + 9), splitness certificate: geometric_involution over Qbar (the geometric automorphism group has order 12 (7 involutions), so the curve is bielliptic over Qbar). Submitted by F. Najman; the curve is due to the database cited under Reference.", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "0", "1", "-4", "-13", "-9", "-1" ], "h": [ "0" ], "claimed_group": [ 2, 6 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "784.b.1" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_2.6.json" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-17 06:12:10 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "1.20000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.6.c.log", "conductor_log": "data/logs/2.6.c.conductor.log", "conductor_cputime_seconds": 0.33 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.8.a", "group": [ 2, 8 ], "group_key": "2.8", "group_bracket": "[2,8]", "order": 16, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "0", "-2", "-4", "3", "8" ], "h": [ "1", "1" ], "f_str": "8*x^5 + 3*x^4 - 4*x^3 - 2*x^2", "h_str": "x + 1", "g": [ "1", "2", "-7", "-16", "12", "32" ], "g_str": "32*x^5 + 12*x^4 - 16*x^3 - 7*x^2 + 2*x + 1", "g_scale": "1", "discriminant": "-118784", "discriminant_digits": 6, "igusa_clebsch": [ "5440", "-336320", "-642023936", "-121634816" ], "g2_invariants": [ "-141985700000/29", "-6991813125/29", "-533176100/29" ], "rational_weierstrass_points": 3, "minimal_model": { "f": [ "-3", "2", "3", "-3", "-1", "1" ], "h": [ "0", "1", "0", "1" ] }, "minimal_discriminant": "-29696", "minimal_discriminant_factored": "-1 * 2^10 * 29", "conductor": "464", "conductor_factored": "2^4 * 29", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "464.a.29696.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/464/a/29696/1", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 8 ], "order": 16, "generators": [ { "order": 2, "mumford": [ "x - 1/2", "0", "1" ] }, { "order": 8, "mumford": [ "x^2 - 1/8*x - 1/4", "1/2*x", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 16, "point_count_primes": [ 3, 5, 7, 11, 13, 17, 19, 23, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 2, 8 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 3, "chi": "T^4 + T^3 + 2*T^2 + 3*T + 9", "primes_tried_below": 500, "admissible_primes": 1, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 + T^3 + 2*T^2 + 3*T + 9" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 44, "reducible_chi": 14 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 41 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "0", "-2", "-4", "3", "8" ], "h": [ "1", "1" ], "claimed_group": [ 2, 8 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "464.a.29696.1" }, "source": { "kind": "import", "name": "paper-t1-041", "file": "table1.txt", "line": 47, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:43 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.320000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.8.a.log", "conductor_log": "data/logs/2.8.a.conductor.log", "conductor_cputime_seconds": 0.2 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.8.b", "group": [ 2, 8 ], "group_key": "2.8", "group_bracket": "[2,8]", "order": 16, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-214", "465", "-279", "447", "-305", "60", "-175" ], "h": [ "1", "0", "1" ], "f_str": "-175*x^6 + 60*x^5 - 305*x^4 + 447*x^3 - 279*x^2 + 465*x - 214", "h_str": "x^2 + 1", "g": [ "-855", "1860", "-1114", "1788", "-1219", "240", "-700" ], "g_str": "-700*x^6 + 240*x^5 - 1219*x^4 + 1788*x^3 - 1114*x^2 + 1860*x - 855", "g_scale": "1", "discriminant": "1095363939978431242017649459200000", "discriminant_digits": 34, "igusa_clebsch": [ "-128329792", "1101430074347584", "-34096508113765752980992", "1095363939978431242017649459200000" ], "g2_invariants": [ "-259314958023705053365123465287944/65288778542186691881278125", "67320092493903554368135318459/5803446981527705945002500", "-46012277421517443109823744/32241372119598366361125" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "-214", "465", "-279", "447", "-305", "60", "-175" ], "h": [ "1", "0", "1" ] }, "minimal_discriminant": "267422836908796689945715200000", "minimal_discriminant_factored": "2^12 * 3^10 * 5^5 * 29^12", "conductor": "225", "conductor_factored": "15 * 15 (product of the elliptic conductors)", "conductor_note": "Magma's genus-2 Conductor gave 230400 (Ogg's formula at 2, no correctness guarantee); the rigorous value is N(E1) N(E2) = 15 * 15 for the elliptic curves ['1', '1', '1', '-135', '-660'] and ['1', '1', '1', '-10', '-10'] to whose product J is isogenous over Q", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "230400" }, "lmfdb": { "label": "225.a.8", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-214%2C465%2C-279%2C447%2C-305%2C60%2C-175%5D%2C%5B1%2C0%2C1%5D%5D", "snapshot_label": "225.a.8" }, "torsion": { "invariants": [ 2, 8 ], "order": 16, "generators": [ { "order": 2, "mumford": [ "x^2 - 54/35*x + 19/35", "0", "2" ] }, { "order": 8, "mumford": [ "x^2 + 7/16*x + 81/80", "-2409/128*x - 15087/640", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 64, "point_count_primes": [ 11, 13, 17, 19, 23, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113 ], "claimed": [ 2, 8 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 90, "reducible_chi": 90 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x - 30/11*z : 8364/1331*x^3 + 14444/1331*x^2*z + 11526/1331*x*z^2 + 24389/1331*y + 17810/1331*z^3 : -24/11*x - z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "1", "1", "-135", "-660" ], [ "1", "1", "1", "-10", "-10" ] ], "conductors": [ "15", "15" ], "conductor": "225", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 41, "reducible_chi": 41 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 36 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-214", "465", "-279", "447", "-305", "60", "-175" ], "h": [ "1", "0", "1" ], "claimed_group": [ 2, 8 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "225.a.8" }, "source": { "kind": "import", "name": "paper-t2-036", "file": "table2.txt", "line": 42, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:37:39 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.16000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.8.b.log", "conductor_log": "data/logs/2.8.b.conductor.log", "conductor_cputime_seconds": 0.22 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "2.8.c", "group": [ 2, 8 ], "group_key": "2.8", "group_bracket": "[2,8]", "order": 16, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-8152375", "2562175", "73961", "83792", "-1345", "847", "49" ], "h": [ "0", "1", "1" ], "f_str": "49*x^6 + 847*x^5 - 1345*x^4 + 83792*x^3 + 73961*x^2 + 2562175*x - 8152375", "h_str": "x^2 + x", "g": [ "-32609500", "10248700", "295845", "335170", "-5379", "3388", "196" ], "g_str": "196*x^6 + 3388*x^5 - 5379*x^4 + 335170*x^3 + 295845*x^2 + 10248700*x - 32609500", "g_scale": "1", "discriminant": "272146120197094257036362228166755678384031793152000000000", "discriminant_digits": 57, "igusa_clebsch": [ "3622349881480", "-148027277018188197419900", "44786356643204846661124668500933000", "272146120197094257036362228166755678384031793152000000000" ], "g2_invariants": [ "1458014179975055909040731135608980159407/5089813862644463489414426787840000", "16478898611653475369394595272347773/1169534435350290323854417920000", "-74180088303510832860855223/423204621406861656096000" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "-8152375", "2562175", "73961", "83792", "-1345", "847", "49" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "66441923876243715096768122111024335542976512000000000", "minimal_discriminant_factored": "2^24 * 3^4 * 5^9 * 7^16 * 11^15 * 17^5 * 127", "conductor": "979254391500", "conductor_factored": "2^2 * 3^2 * 5^3 * 7^2 * 11^2 * 17^2 * 127", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-8152375%2C2562175%2C73961%2C83792%2C-1345%2C847%2C49%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 8 ], "order": 16, "generators": [ { "order": 2, "mumford": [ "x^2 + 275/14*x - 55", "0", "2" ] }, { "order": 8, "mumford": [ "x^2 + 121/7*x - 110", "5456/7*x + 4840", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "the submitted generators generate a subgroup of order 16, but the point-count bound is 32; TorsionSubgroup was run", "generators_submitted": 2, "point_count_gcd": 32, "point_count_primes": [ 13, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 131 ], "claimed": [ 2, 8 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 88, "reducible_chi": 57 }, "q_simple": { "certified": true, "prime": 19, "chi": "T^4 - 10*T^2 + 361" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "richelot", "over": "Q", "detail": "(2,2)-isogenous over Q to a Weil restriction of an elliptic curve E over Q(sqrt(-55)) (the surface is E x E^sigma)" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 39, "reducible_chi": 27 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "F. Najman", "year": 2026, "historical": false }, "reference": "new", "sources": [], "rm": false, "notes": "First Q-simple, geometrically split example of [2,8] for the census (the gsplit cell is empty). Construction: quadratic-conjugate (2,2)-gluing. E = E_{2,8}(t) (Tate normal form of HLP Table 3) with t = (-1 + sqrt(-55))/8 over K = Q(sqrt(-55)), E(K)_tors = Z/2 x Z/8 =

x ; E is glued with its Galois conjugate E^sigma along the G_K-isomorphism psi: E[2] -> E^sigma[2] that fixes the 2-torsion point 4P and swaps the other two (psi^sigma psi = id, so the glued abelian surface Res_{K/Q}(E)/graph(psi) and the curve descend to Q). The Kummer class of P + T is twisted-invariant, which gives the extra factor 2 (J(Q)_tors = [2,8], against [2,4] for a generic member of the family). E is not a Q-curve (a_P(E) <> a_P(E^sigma) at 15 of 34 split primes below 400), so J is simple over Q (also certified by chi_19 irreducible); J is (2,2)-isogenous over Q to Res_{K/Q} E, which Magma's Richelot search finds. Found by a search over t = t1 + sqrt(s) with a Kummer-class prefilter (torsion_jac code/gsplit_gain_search.m). Conductor 979254391500 = 2^2 * 3^2 * 5^3 * 7^2 * 11^2 * 17^2 * 127 (Magma's Conductor on the reduced minimal model; it agrees with the rigorous value below). A second example with the same construction: t = -2 + sqrt(10)/4 over Q(sqrt 10), y^2 = 1521x^6 + 4368x^5 - 20264x^4 - 17376x^3 - 102569x^2 - 722580x + 906724, conductor 10145386368000. Rigorous conductor: J is (2,2)-isogenous over Q to Res_(K/Q) E with K the quadratic field of discriminant -55, so N(J) = |d_K|^2 * Norm(f_E) = 3025 * 323720460 = 979254391500 (computed with Magma; conductor is an isogeny invariant).", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "-8152375", "2562175", "73961", "83792", "-1345", "847", "49" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 8 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_2.8.json" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-17 06:12:13 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "1.60000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.8.c.log", "conductor_log": "data/logs/2.8.c.conductor.log", "conductor_cputime_seconds": 0.63 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.10.a", "group": [ 2, 10 ], "group_key": "2.10", "group_bracket": "[2,10]", "order": 20, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "1", "1", "-4", "-2", "3" ], "h": [ "1", "1" ], "f_str": "3*x^5 - 2*x^4 - 4*x^3 + x^2 + x", "h_str": "x + 1", "g": [ "1", "6", "5", "-16", "-8", "12" ], "g_str": "12*x^5 - 8*x^4 - 16*x^3 + 5*x^2 + 6*x + 1", "g_scale": "1", "discriminant": "236800", "discriminant_digits": 6, "igusa_clebsch": [ "5056", "289984", "439647168", "34099200" ], "g2_invariants": [ "100828984082432/8325", "3438682756096/8325", "140342016064/8325" ], "rational_weierstrass_points": 3, "minimal_model": { "f": [ "0", "-3", "-2", "4", "1", "-2" ], "h": [ "0", "0", "1", "1" ] }, "minimal_discriminant": "8325", "minimal_discriminant_factored": "3^2 * 5^2 * 37", "conductor": "555", "conductor_factored": "3 * 5 * 37", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "555.a.8325.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/555/a/8325/1", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 10 ], "order": 20, "generators": [ { "order": 2, "mumford": [ "x^2 - 2/3*x - 1/3", "0", "2" ] }, { "order": 10, "mumford": [ "x", "-1", "1" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 20, "point_count_primes": [ 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 2, 10 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 7, "chi": "T^4 - T^3 - 2*T^2 - 7*T + 49", "primes_tried_below": 500, "admissible_primes": 1, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 7, "chi": "T^4 - T^3 - 2*T^2 - 7*T + 49" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 42, "reducible_chi": 9 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 42 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "1", "1", "-4", "-2", "3" ], "h": [ "1", "1" ], "claimed_group": [ 2, 10 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "555.a.8325.1" }, "source": { "kind": "import", "name": "paper-t1-042", "file": "table1.txt", "line": 48, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:43 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.320000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.10.a.log", "conductor_log": "data/logs/2.10.a.conductor.log", "conductor_cputime_seconds": 0.14 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.10.b", "group": [ 2, 10 ], "group_key": "2.10", "group_bracket": "[2,10]", "order": 20, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "0", "0", "-1", "-1", "-1", "-1" ], "h": [ "1", "1", "1", "1" ], "f_str": "-x^5 - x^4 - x^3 - x^2", "h_str": "x^3 + x^2 + x + 1", "g": [ "1", "2", "-1", "0", "-1", "-2", "1" ], "g_str": "x^6 - 2*x^5 - x^4 - x^2 + 2*x + 1", "g_scale": "1", "discriminant": "-2097152", "discriminant_digits": 7, "igusa_clebsch": [ "-416", "-512", "-163840", "-2097152" ], "g2_invariants": [ "742586", "129623/4", "-1521/8" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "0", "0", "-1", "-1", "-1", "-1" ], "h": [ "1", "1", "1", "1" ] }, "minimal_discriminant": "-512", "minimal_discriminant_factored": "-1 * 2^9", "conductor": "256", "conductor_factored": "2^8", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "256.a.512.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/256/a/512/1", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 10 ], "order": 20, "generators": [ { "order": 2, "mumford": [ "x^2 - 1", "0", "2" ] }, { "order": 10, "mumford": [ "x - 1", "-x^3 + 1", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 20, "point_count_primes": [ 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101 ], "claimed": [ 2, 10 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 94, "reducible_chi": 33 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 + 2*T^3 + 2*T^2 + 6*T + 9" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "geometric_involution", "over": "Qbar", "detail": "the geometric automorphism group has order 8 (5 involutions), so the curve is bielliptic over Qbar" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 45, "reducible_chi": 16 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Ogg 1973", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 37 (source: \\cite{Ogg1973})", "sources": [ "Ogg1973" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "0", "-1", "-1", "-1", "-1" ], "h": [ "1", "1", "1", "1" ], "claimed_group": [ 2, 10 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "256.a.512.1" }, "source": { "kind": "import", "name": "paper-t2-037", "file": "table2.txt", "line": 43, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:37:40 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.13000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.10.b.log", "conductor_log": "data/logs/2.10.b.conductor.log", "conductor_cputime_seconds": 0.0700001 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.10.c", "group": [ 2, 10 ], "group_key": "2.10", "group_bracket": "[2,10]", "order": 20, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "0", "2", "4", "2", "0", "1" ], "h": [ "1", "0", "1" ], "f_str": "x^5 + 2*x^3 + 4*x^2 + 2*x", "h_str": "x^2 + 1", "g": [ "1", "8", "18", "8", "1", "4" ], "g_str": "4*x^5 + x^4 + 8*x^3 + 18*x^2 + 8*x + 1", "g_scale": "1", "discriminant": "-3066624", "discriminant_digits": 7, "igusa_clebsch": [ "1376", "-49088", "-33691712", "-49065984" ], "g2_invariants": [ "-150536645632/11979", "-8874253312/11979", "-1356160144/11979" ], "rational_weierstrass_points": 3, "minimal_model": { "f": [ "0", "-1", "0", "-2", "4", "-2" ], "h": [ "0", "1", "0", "1" ] }, "minimal_discriminant": "-11979", "minimal_discriminant_factored": "-1 * 3^2 * 11^3", "conductor": "363", "conductor_factored": "3 * 11^2", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "363.a.11979.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/363/a/11979/1", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 10 ], "order": 20, "generators": [ { "order": 2, "mumford": [ "x + 1", "0", "1" ] }, { "order": 10, "mumford": [ "x^2 - 5*x - 2", "-25*x - 9", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "the submitted generators generate a subgroup of order 20, but the point-count bound is 40; TorsionSubgroup was run", "generators_submitted": 2, "point_count_gcd": 40, "point_count_primes": [ 5, 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 2, 10 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "cover", "over": "Q", "detail": "a nonconstant map of degree 3 from y^2 = g to the genus-1 curve v^2 = (1)u^3 + (5)u^2 + (3)u + (7) over Q, (x,y) -> (p/q, y*hh/q^2) with p = 4*x^3 + x + 1, q = 4*x^2 + 5*x + 1, hh = 8*x^3 + 24*x^2 + 16*x + 4; degree-3 map from the even model y^2 = 4x^5 + x^4 + 8x^3 + 18x^2 + 8x + 1 onto v^2 = u^3 + 5u^2 + 3u + 7 (the elliptic curve 11a3), computed for this submission" }, "isogenous_product": null, "cover_submitted": true, "cover_accepted": true, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "LMFDB 363.a.11979.1, https://www.lmfdb.org/Genus2Curve/Q/363/a/11979/1", "sources": [], "rm": false, "notes": "Q-split witness: the census has only geometrically simple and Q-simple (geometrically split) curves with this torsion group. Not a new curve. LMFDB: torsion [2,10], not simple over Q (is_simple_base = false); J ~ 11a x 33a over Q, glued along their 3-torsion (a_p(11a) = a_p(33a) mod 3). The cover above is a degree-3 map u = (4x^3 + x + 1)/((4x + 1)(x + 1)) onto 11a3, found by solving for the ramification of a degree-3 elliptic subcover whose fibre over the origin is the three Weierstrass points over the roots of (4x + 1)(x + 1) and infinity. Output of pipeline/magma/prepare_submission.m (Magma V2.29-4): J(Q)_tors = [2,10]; gcd of #J(F_p) over 25 good primes = 40 (not sharp); no strict prime below 500; conductor 363 = 3 * 11^2.", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "0", "2", "4", "2", "0", "1" ], "h": [ "1", "0", "1" ], "claimed_group": [ 2, 10 ], "claimed_class": "split", "cover": { "field": "", "cremona": "", "cubic": [ "1", "5", "3", "7" ], "p": [ "1", "1", "0", "4" ], "q": [ "1", "5", "4" ], "hh": [ "4", "16", "24", "8" ], "note": "degree-3 map from the even model y^2 = 4x^5 + x^4 + 8x^3 + 18x^2 + 8x + 1 onto v^2 = u^3 + 5u^2 + 3u + 7 (the elliptic curve 11a3), computed for this submission" }, "lmfdb_label_given": "" }, "source": { "kind": "issue", "number": 6, "url": "https://github.com/Genus-2-torsion/Genus-2-torsion.github.io/issues/6" }, "dates": { "submitted": "2026-10-14", "verified": "2026-10-14 10:49:17 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.43000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.10.c.log", "conductor_log": "data/logs/2.10.c.conductor.log", "conductor_cputime_seconds": 0.16 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.12.a", "group": [ 2, 12 ], "group_key": "2.12", "group_bracket": "[2,12]", "order": 24, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "-1", "2", "14", "-8", "1" ], "h": [ "0", "1", "1" ], "f_str": "x^5 - 8*x^4 + 14*x^3 + 2*x^2 - x", "h_str": "x^2 + x", "g": [ "0", "-4", "9", "58", "-31", "4" ], "g_str": "4*x^5 - 31*x^4 + 58*x^3 + 9*x^2 - 4*x", "g_scale": "1", "discriminant": "21067776", "discriminant_digits": 8, "igusa_clebsch": [ "24008", "820996", "6480164616", "337084416" ], "g2_invariants": [ "243405270090015001/82296", "4955375073324349/41148", "65790314289164/10287" ], "rational_weierstrass_points": 3, "minimal_model": { "f": [ "0", "-1", "2", "14", "-8", "1" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "82296", "minimal_discriminant_factored": "2^3 * 3^4 * 127", "conductor": "762", "conductor_factored": "2 * 3 * 127", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "762.a.82296.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/762/a/82296/1", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 12 ], "order": 24, "generators": [ { "order": 2, "mumford": [ "x", "0", "1" ] }, { "order": 12, "mumford": [ "x - 1", "6", "1" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 24, "point_count_primes": [ 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 2, 12 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 7, "chi": "T^4 - T^3 + 6*T^2 - 7*T + 49", "primes_tried_below": 500, "admissible_primes": 2, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 5, "chi": "T^4 + 3*T^3 + 4*T^2 + 15*T + 25" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 43, "reducible_chi": 22 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 43 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "-1", "2", "14", "-8", "1" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 12 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "762.a.82296.1" }, "source": { "kind": "import", "name": "paper-t1-043", "file": "table1.txt", "line": 49, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:44 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.270000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.12.a.log", "conductor_log": "data/logs/2.12.a.conductor.log", "conductor_cputime_seconds": 0.13 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.12.b", "group": [ 2, 12 ], "group_key": "2.12", "group_bracket": "[2,12]", "order": 24, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-5", "15", "-8", "-10", "4", "2" ], "h": [ "0", "1", "1", "1" ], "f_str": "2*x^5 + 4*x^4 - 10*x^3 - 8*x^2 + 15*x - 5", "h_str": "x^3 + x^2 + x", "g": [ "-20", "60", "-31", "-38", "19", "10", "1" ], "g_str": "x^6 + 10*x^5 + 19*x^4 - 38*x^3 - 31*x^2 + 60*x - 20", "g_scale": "1", "discriminant": "149299200", "discriminant_digits": 9, "igusa_clebsch": [ "46888", "848356", "13017438568", "149299200" ], "g2_invariants": [ "6916057684302385301/36450", "5303516319500302/675", "1294426477922/3" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "0", "4", "21", "39", "21", "4" ], "h": [ "1", "0", "0", "1" ] }, "minimal_discriminant": "36450", "minimal_discriminant_factored": "2 * 3^6 * 5^2", "conductor": "450", "conductor_factored": "15 * 30 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 15 * 30 for the elliptic curves ['1', '1', '1', '-5', '2'] and ['1', '0', '1', '-19', '26'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "450" }, "lmfdb": { "label": "450.a.36450.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/450/a/36450/1", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 12 ], "order": 24, "generators": [ { "order": 2, "mumford": [ "x^2 + 4*x - 4", "0", "2" ] }, { "order": 12, "mumford": [ "1", "-x^3 - 5*x^2", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 96, "point_count_primes": [ 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 2, 12 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x : -x^3 + 2*x^2*z + y : x - z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "1", "1", "-5", "2" ], [ "1", "0", "1", "-19", "26" ] ], "conductors": [ "15", "30" ], "conductor": "450", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 38 (source: \\cite{HLP2000})", "sources": [ "HLP2000" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-5", "15", "-8", "-10", "4", "2" ], "h": [ "0", "1", "1", "1" ], "claimed_group": [ 2, 12 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "450.a.36450.1" }, "source": { "kind": "import", "name": "paper-t2-038", "file": "table2.txt", "line": 44, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:37:41 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.17000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.12.b.log", "conductor_log": "data/logs/2.12.b.conductor.log", "conductor_cputime_seconds": 0.2 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "2.14.a", "group": [ 2, 14 ], "group_key": "2.14", "group_bracket": "[2,14]", "order": 28, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "1", "1", "0", "-1", "-2" ], "h": [ "0", "1", "0", "1" ], "f_str": "-2*x^4 - x^3 + x + 1", "h_str": "x^3 + x", "g": [ "4", "4", "1", "-4", "-6", "0", "1" ], "g_str": "x^6 - 6*x^4 - 4*x^3 + x^2 + 4*x + 4", "g_scale": "1", "discriminant": "-556793856", "discriminant_digits": 9, "igusa_clebsch": [ "-768", "-1536", "-5802240", "-556793856" ], "g2_invariants": [ "3538944/59", "153600/59", "-33808/59" ], "rational_weierstrass_points": 3, "minimal_model": { "f": [ "1", "1", "0", "-1", "-2" ], "h": [ "0", "1", "0", "1" ] }, "minimal_discriminant": "-135936", "minimal_discriminant_factored": "-1 * 2^8 * 3^2 * 59", "conductor": "1416", "conductor_factored": "2^3 * 3 * 59", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "1416.b.135936.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/1416/b/135936/1", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 14 ], "order": 28, "generators": [ { "order": 2, "mumford": [ "x^2 + 3*x + 2", "0", "2" ] }, { "order": 14, "mumford": [ "x^2 - x", "-2*x + 2", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 28, "point_count_primes": [ 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 2, 14 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 17, "chi": "T^4 - 4*T^3 + 6*T^2 - 68*T + 289", 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"lmfdb_label_given": "1416.b.135936.1" }, "source": { "kind": "import", "name": "paper-t1-044", "file": "table1.txt", "line": 50, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:44 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.300000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.14.a.log", "conductor_log": "data/logs/2.14.a.conductor.log", "conductor_cputime_seconds": 0.14 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.14.b", "group": [ 2, 14 ], "group_key": "2.14", "group_bracket": "[2,14]", "order": 28, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new 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"url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B10%2C-6%2C-24%2C-5%2C-9%2C-3%2C9%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "86436.z.3" }, "torsion": { "invariants": [ 2, 14 ], "order": 28, "generators": [ { "order": 2, "mumford": [ "x^2 - 2/3*x - 5/3", "0", "2" ] }, { "order": 14, "mumford": [ "x - 1/2", "6*x^3 - 3/4", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 28, "point_count_primes": [ 5, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 2, 14 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "cover", "over": "Q", "detail": "a nonconstant map of degree 3 from y^2 = g to the genus-1 curve v^2 = (1)u^3 + (0)u^2 + (-182763)u + (31201254) over Q (the elliptic curve 294b2), (x,y) -> (p/q, y*hh/q^2) with p = 435*x^3 - 367/2*x^2 - 148*x + 509/2, q = x^3 - 7/6*x^2 - 4/3*x + 5/6, hh = 972*x^3 - 324*x^2 - 216*x - 216; degree-3 map to the elliptic curve 294b2 over Q [Genus2Torsion/verify_split_certificates.m, cover 13]" }, "isogenous_product": null, "cover_submitted": true, "cover_accepted": true, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 39 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "10", "-6", "-24", "-5", "-9", "-3", "9" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 14 ], "claimed_class": "split", "cover": { "field": "", "cremona": "294b2", "cubic": [], "p": [ "509/2", "-148", "-367/2", "435" ], "q": [ "5/6", "-4/3", "-7/6", "1" ], "hh": [ "-216", "-216", "-324", "972" ], "note": "degree-3 map to the elliptic curve 294b2 over Q [Genus2Torsion/verify_split_certificates.m, cover 13]" }, "lmfdb_label_given": "86436.z.3" }, "source": { "kind": "import", "name": "paper-t2-039", "file": "table2.txt", "line": 45, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:37:43 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.19000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.14.b.log", "conductor_log": "data/logs/2.14.b.conductor.log", "conductor_cputime_seconds": 0.35 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.16.a", "group": [ 2, 16 ], "group_key": "2.16", "group_bracket": "[2,16]", "order": 32, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "8", "4", "-3", "1", "-1", "-1" ], "h": [ "0", "0", "1", "1" ], "f_str": "-x^5 - x^4 + x^3 - 3*x^2 + 4*x + 8", "h_str": "x^3 + x^2", "g": [ "32", "16", "-12", "4", "-3", "-2", "1" ], "g_str": "x^6 - 2*x^5 - 3*x^4 + 4*x^3 - 12*x^2 + 16*x + 32", "g_scale": "1", "discriminant": "-746066018304", "discriminant_digits": 12, "igusa_clebsch": [ "-9440", "888640", "-3988585984", "-746066018304" ], "g2_invariants": [ "2234138434375/177876", "156473125625/355752", "-474199225/711504" ], "rational_weierstrass_points": 3, "minimal_model": { "f": [ "8", "4", "-3", "1", "-1", "-1" ], "h": [ "0", "0", "1", "1" ] }, "minimal_discriminant": "-182145024", "minimal_discriminant_factored": "-1 * 2^12 * 3^6 * 61", "conductor": "4392", "conductor_factored": "2^3 * 3^2 * 61", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "4392.a.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B8%2C4%2C-3%2C1%2C-1%2C-1%5D%2C%5B0%2C0%2C1%2C1%5D%5D", "snapshot_label": "4392.a.1" }, "torsion": { "invariants": [ 2, 16 ], "order": 32, "generators": [ { "order": 2, "mumford": [ "x^2 + 3*x + 2", "0", "2" ] }, { "order": 16, "mumford": [ "x^2 - 2*x - 2", "2*x - 2", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 32, "point_count_primes": [ 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 2, 16 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 5, "chi": "T^4 + T^3 + 5*T + 25", "primes_tried_below": 500, "admissible_primes": 1, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 5, "chi": "T^4 + T^3 + 5*T + 25" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 43, "reducible_chi": 5 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 45 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "8", "4", "-3", "1", "-1", "-1" ], "h": [ "0", "0", "1", "1" ], "claimed_group": [ 2, 16 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "4392.a.1" }, "source": { "kind": "import", "name": "paper-t1-045", "file": "table1.txt", "line": 51, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:45 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.250000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.16.a.log", "conductor_log": "data/logs/2.16.a.conductor.log", "conductor_cputime_seconds": 0.16 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.16.b", "group": [ 2, 16 ], "group_key": "2.16", "group_bracket": "[2,16]", "order": 32, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "2", "7", "16", "18", "16", "7", "2" ], "h": [ "1", "1", "1", "1" ], "f_str": "2*x^6 + 7*x^5 + 16*x^4 + 18*x^3 + 16*x^2 + 7*x + 2", "h_str": "x^3 + x^2 + x + 1", "g": [ "9", "30", "67", "76", "67", "30", "9" ], "g_str": "9*x^6 + 30*x^5 + 67*x^4 + 76*x^3 + 67*x^2 + 30*x + 9", "g_scale": "1", "discriminant": "-7925422620672", "discriminant_digits": 13, "igusa_clebsch": [ "-20608", "3595264", "-23512416256", "-7925422620672" ], "g2_invariants": [ "3461619737632/59049", "13855292920/6561", "63869344/729" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "2", "-8", "16", "-20", "16", "-8", "2" ], "h": [ "1", "1", "1", "1" ] }, "minimal_discriminant": "-1934917632", "minimal_discriminant_factored": "-1 * 2^15 * 3^10", "conductor": "4608", "conductor_factored": "48 * 96 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 48 * 96 for the elliptic curves ['0', '1', '0', '16', '180'] and ['0', '1', '0', '8', '8'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "4608" }, "lmfdb": { "label": "4608.h.5", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B2%2C7%2C16%2C18%2C16%2C7%2C2%5D%2C%5B1%2C1%2C1%2C1%5D%5D", "snapshot_label": "4608.h.5" }, "torsion": { "invariants": [ 2, 16 ], "order": 32, "generators": [ { "order": 2, "mumford": [ "x^2 + 2/3*x + 1/3", "0", "2" ] }, { "order": 16, "mumford": [ "x^2 + x", "7*x + 3", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 32, "point_count_primes": [ 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 2, 16 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 93, "reducible_chi": 93 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (z : y : x) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "0", "1", "0", "16", "180" ], [ "0", "1", "0", "8", "8" ] ], "conductors": [ "48", "96" ], "conductor": "4608", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 44, "reducible_chi": 44 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 40 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "2", "7", "16", "18", "16", "7", "2" ], "h": [ "1", "1", "1", "1" ], "claimed_group": [ 2, 16 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "4608.h.5" }, "source": { "kind": "import", "name": "paper-t2-040", "file": "table2.txt", "line": 46, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:37:45 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.19000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.16.b.log", "conductor_log": "data/logs/2.16.b.conductor.log", "conductor_cputime_seconds": 0.14 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.18.a", "group": [ 2, 18 ], "group_key": "2.18", "group_bracket": "[2,18]", "order": 36, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "7", "2", "1", "3", "-5", "-1", "1" ], "h": [ "0", "1", "1" ], "f_str": "x^6 - x^5 - 5*x^4 + 3*x^3 + x^2 + 2*x + 7", "h_str": "x^2 + x", "g": [ "28", "8", "5", "14", "-19", "-4", "4" ], "g_str": "4*x^6 - 4*x^5 - 19*x^4 + 14*x^3 + 5*x^2 + 8*x + 28", "g_scale": "1", "discriminant": "-77110337470464", "discriminant_digits": 14, "igusa_clebsch": [ "-25464", "18256356", "-234179021592", "-77110337470464" ], "g2_invariants": [ "1344549883801301/77472288", "30785401760275/77472288", "-28099121881/318816" ], "rational_weierstrass_points": 3, "minimal_model": { "f": [ "7", "2", "1", "3", "-5", "-1", "1" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "-18825765984", "minimal_discriminant_factored": "-1 * 2^5 * 3^15 * 41", "conductor": "4428", "conductor_factored": "2^2 * 3^3 * 41", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "4428.a.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B7%2C2%2C1%2C3%2C-5%2C-1%2C1%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "4428.a.2" }, "torsion": { "invariants": [ 2, 18 ], "order": 36, "generators": [ { "order": 2, "mumford": [ "x^2 - 4", "0", "2" ] }, { "order": 18, "mumford": [ "x + 1", "2*x^3 + 2", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 36, "point_count_primes": [ 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 2, 18 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 17, "chi": "T^4 + T^3 + 16*T^2 + 17*T + 289", "primes_tried_below": 500, "admissible_primes": 5, "reducible_chi": 3 }, "q_simple": { "certified": true, "prime": 13, "chi": "T^4 - T^3 - 12*T^2 - 13*T + 169" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 43, "reducible_chi": 17 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 46 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "7", "2", "1", "3", "-5", "-1", "1" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 18 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "4428.a.2" }, "source": { "kind": "import", "name": "paper-t1-046", "file": "table1.txt", "line": 52, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:46 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.320000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.18.a.log", "conductor_log": "data/logs/2.18.a.conductor.log", "conductor_cputime_seconds": 0.19 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.18.b", "group": [ 2, 18 ], "group_key": "2.18", "group_bracket": "[2,18]", "order": 36, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "36", "66", "12", "-5", "-3", "-3", "1" ], "h": [ "0", "1", "1" ], "f_str": "x^6 - 3*x^5 - 3*x^4 - 5*x^3 + 12*x^2 + 66*x + 36", "h_str": "x^2 + x", "g": [ "144", "264", "49", "-18", "-11", "-12", "4" ], "g_str": "4*x^6 - 12*x^5 - 11*x^4 - 18*x^3 + 49*x^2 + 264*x + 144", "g_scale": "1", "discriminant": "-429514854361989120", "discriminant_digits": 18, "igusa_clebsch": [ "-254392", "536068420", "-82604216222008", "-429514854361989120" ], "g2_invariants": [ "32513768825556207758999/104862024990720", "24483098474191432777/2184625520640", "-319478150366747/1685667840" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "36", "66", "12", "-5", "-3", "-3", "1" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "-104862024990720", "minimal_discriminant_factored": "-1 * 2^22 * 3^6 * 5 * 19^3", "conductor": "64980", "conductor_factored": "2^2 * 3^2 * 5 * 19^2", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "64980.bo.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B36%2C66%2C12%2C-5%2C-3%2C-3%2C1%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "64980.bo.1" }, "torsion": { "invariants": [ 2, 18 ], "order": 36, "generators": [ { "order": 2, "mumford": [ "x^2 - 5/2*x - 2", "0", "2" ] }, { "order": 18, "mumford": [ "x^2 + 3/2*x", "-8*x - 12", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 36, "point_count_primes": [ 7, 11, 13, 17, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 2, 18 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 91, "reducible_chi": 91 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "cover", "over": "Q", "detail": "a nonconstant map of degree 3 from y^2 = g to the genus-1 curve v^2 = (1)u^3 + (0)u^2 + (-10395)u + (31158) over Q (the elliptic curve 114a1), (x,y) -> (p/q, y*hh/q^2) with p = 147*x^3 - 615/2*x^2 + 156*x - 1656, q = x^3 - 13/2*x^2 + 4*x + 24, hh = 648*x^3 - 648*x^2 - 15552*x + 31104; degree-3 map to the elliptic curve 114a1 over Q [Genus2Torsion/verify_split_certificates.m, cover 15]" }, "isogenous_product": null, "cover_submitted": true, "cover_accepted": true, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 42, "reducible_chi": 42 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 41 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "36", "66", "12", "-5", "-3", "-3", "1" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 18 ], "claimed_class": "split", "cover": { "field": "", "cremona": "114a1", "cubic": [], "p": [ "-1656", "156", "-615/2", "147" ], "q": [ "24", "4", "-13/2", "1" ], "hh": [ "31104", "-15552", "-648", "648" ], "note": "degree-3 map to the elliptic curve 114a1 over Q [Genus2Torsion/verify_split_certificates.m, cover 15]" }, "lmfdb_label_given": "64980.bo.1" }, "source": { "kind": "import", "name": "paper-t2-041", "file": "table2.txt", "line": 47, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:37:46 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.33000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.18.b.log", "conductor_log": "data/logs/2.18.b.conductor.log", "conductor_cputime_seconds": 0.32 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.20.a", "group": [ 2, 20 ], "group_key": "2.20", "group_bracket": "[2,20]", "order": 40, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "-180", "-251", "185", "28", "-30", "4" ], "h": [ "0", "0", "1" ], "f_str": "4*x^6 - 30*x^5 + 28*x^4 + 185*x^3 - 251*x^2 - 180*x", "h_str": "x^2", "g": [ "0", "-720", "-1004", "740", "113", "-120", "16" ], "g_str": "16*x^6 - 120*x^5 + 113*x^4 + 740*x^3 - 1004*x^2 - 720*x", "g_scale": "1", "discriminant": "72761549311180800000000", "discriminant_digits": 23, "igusa_clebsch": [ "8556832", "237441772864", "693009255798630912", "72761549311180800000000" ], "g2_invariants": [ "1367146752883583423268937001/17347705200000000", "4320702761137577462162957/1387816416000000", "16905683842637790410401/111025313280000" ], "rational_weierstrass_points": 3, "minimal_model": { "f": [ "-244", "-140", "233", "78", "-62", "-6", "4" ], "h": [ "1", "0", "1" ] }, "minimal_discriminant": "17764050124800000000", "minimal_discriminant_factored": "2^20 * 3^4 * 5^8 * 7^4 * 223", "conductor": "936600", "conductor_factored": "2^3 * 3 * 5^2 * 7 * 223", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "936600.b.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C-180%2C-251%2C185%2C28%2C-30%2C4%5D%2C%5B0%2C0%2C1%5D%5D", "snapshot_label": "936600.b.1" }, "torsion": { "invariants": [ 2, 20 ], "order": 40, "generators": [ { "order": 2, "mumford": [ "x^2 - 4*x", "0", "2" ] }, { "order": 20, "mumford": [ "x^2 - 15/4*x - 3/8", "-5/2*x + 15/2", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 40, "point_count_primes": [ 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 2, 20 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 17, "chi": "T^4 - 3*T^3 + 4*T^2 - 51*T + 289", "primes_tried_below": 500, "admissible_primes": 3, "reducible_chi": 2 }, "q_simple": { "certified": true, "prime": 17, "chi": "T^4 - 3*T^3 + 4*T^2 - 51*T + 289" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 42, "reducible_chi": 19 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 47 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "-180", "-251", "185", "28", "-30", "4" ], "h": [ "0", "0", "1" ], "claimed_group": [ 2, 20 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "936600.b.1" }, "source": { "kind": "import", "name": "paper-t1-047", "file": "table1.txt", "line": 53, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:46 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.260000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.20.a.log", "conductor_log": "data/logs/2.20.a.conductor.log", "conductor_cputime_seconds": 0.4 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.20.b", "group": [ 2, 20 ], "group_key": "2.20", "group_bracket": "[2,20]", "order": 40, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "0", "-18", "42", "-20", "15", "-3", "1" ], "h": [ "0", "1", "1" ], "f_str": "x^6 - 3*x^5 + 15*x^4 - 20*x^3 + 42*x^2 - 18*x", "h_str": "x^2 + x", "g": [ "0", "-72", "169", "-78", "61", "-12", "4" ], "g_str": "4*x^6 - 12*x^5 + 61*x^4 - 78*x^3 + 169*x^2 - 72*x", "g_scale": "1", "discriminant": "170629491949830144", "discriminant_digits": 18, "igusa_clebsch": [ "-93880", "641545348", "-14522235977080", "170629491949830144" ], "g2_invariants": [ "-222543784726176584375/41657590808064", "28275667993091375/771436866816", "-132842857097375/128572811136" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "0", "-18", "42", "-20", "15", "-3", "1" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "41657590808064", "minimal_discriminant_factored": "2^9 * 3^8 * 7 * 11^6", "conductor": "30492", "conductor_factored": "2^2 * 3^2 * 7 * 11^2", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "30492.q.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C-18%2C42%2C-20%2C15%2C-3%2C1%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "30492.q.2" }, "torsion": { "invariants": [ 2, 20 ], "order": 40, "generators": [ { "order": 2, "mumford": [ "x^2 - 1/2*x", "0", "2" ] }, { "order": 20, "mumford": [ "x^2 - 3/2*x", "11*x", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 40, "point_count_primes": [ 5, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 2, 20 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 91, "reducible_chi": 91 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "cover", "over": "Q", "detail": "a nonconstant map of degree 3 from y^2 = g to the genus-1 curve v^2 = (1)u^3 + (0)u^2 + (-125739)u + (62756262) over Q (the elliptic curve 462f1), (x,y) -> (p/q, y*hh/q^2) with p = -501*x^3 + 2505/2*x^2 - 654*x + 4509/2, q = x^3 - 5/2*x^2 + 10*x - 9/2, hh = 26136*x^2 - 58806*x + 58806; degree-3 map to the elliptic curve 462f1 over Q [Genus2Torsion/verify_split_certificates.m, cover 14]" }, "isogenous_product": null, "cover_submitted": true, "cover_accepted": true, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 42, "reducible_chi": 42 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 42 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "-18", "42", "-20", "15", "-3", "1" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 20 ], "claimed_class": "split", "cover": { "field": "", "cremona": "462f1", "cubic": [], "p": [ "4509/2", "-654", "2505/2", "-501" ], "q": [ "-9/2", "10", "-5/2", "1" ], "hh": [ "58806", "-58806", "26136" ], "note": "degree-3 map to the elliptic curve 462f1 over Q [Genus2Torsion/verify_split_certificates.m, cover 14]" }, "lmfdb_label_given": "30492.q.2" }, "source": { "kind": "import", "name": "paper-t2-042", "file": "table2.txt", "line": 48, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:37:48 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.34000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.20.b.log", "conductor_log": "data/logs/2.20.b.conductor.log", "conductor_cputime_seconds": 0.25 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.22.a", "group": [ 2, 22 ], "group_key": "2.22", "group_bracket": "[2,22]", "order": 44, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "-6", "12", "-5", "9", "-3", "1" ], "h": [ "0", "1", "1" ], "f_str": "x^6 - 3*x^5 + 9*x^4 - 5*x^3 + 12*x^2 - 6*x", "h_str": "x^2 + x", "g": [ "0", "-24", "49", "-18", "37", "-12", "4" ], "g_str": "4*x^6 - 12*x^5 + 37*x^4 - 18*x^3 + 49*x^2 - 24*x", "g_scale": "1", "discriminant": "4716617137127424", "discriminant_digits": 16, "igusa_clebsch": [ "-15544", "31561924", "-136103995576", "4716617137127424" ], "g2_invariants": [ "-27692617238317943/1151517855744", "34938075847741/31986607104", "121959418945/888516864" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "0", "-6", "12", "-5", "9", "-3", "1" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "1151517855744", "minimal_discriminant_factored": "2^12 * 3^12 * 23^2", "conductor": "19044", "conductor_factored": "2^2 * 3^2 * 23^2", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "19044.h.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C-6%2C12%2C-5%2C9%2C-3%2C1%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "19044.h.2" }, "torsion": { "invariants": [ 2, 22 ], "order": 44, "generators": [ { "order": 2, "mumford": [ "x^2 + 1/2*x + 3/2", "0", "2" ] }, { "order": 22, "mumford": [ "x + 1", "-2*x^3 + 10", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 44, "point_count_primes": [ 5, 7, 11, 13, 17, 19, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 2, 22 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 5, "chi": "T^4 + 2*T^3 + 6*T^2 + 10*T + 25", "primes_tried_below": 500, "admissible_primes": 1, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 5, "chi": "T^4 + 2*T^3 + 6*T^2 + 10*T + 25" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 43, "reducible_chi": 5 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 48 (source: \\cite{BookerSutherland}$^{\\rm RM}$)", "sources": [ "BookerSutherland" ], "rm": true, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "-6", "12", "-5", "9", "-3", "1" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 22 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "19044.h.2" }, "source": { "kind": "import", "name": "paper-t1-048", "file": "table1.txt", "line": 54, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:47 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.370000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.22.a.log", "conductor_log": "data/logs/2.22.a.conductor.log", "conductor_cputime_seconds": 0.19 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.24.a", "group": [ 2, 24 ], "group_key": "2.24", "group_bracket": "[2,24]", "order": 48, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "15", "0", "-6", "0", "-1" ], "h": [ "0", "1", "0", "1" ], "f_str": "-x^4 - 6*x^2 + 15", "h_str": "x^3 + x", "g": [ "60", "0", "-23", "0", "-2", "0", "1" ], "g_str": "x^6 - 2*x^4 - 23*x^2 + 60", "g_scale": "1", "discriminant": "-103195607040", "discriminant_digits": 12, "igusa_clebsch": [ "-15136", "3842368", "-22254697984", "-103195607040" ], "g2_invariants": [ "94703427014372/98415", "962361791798/32805", "-597803888/2187" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "15", "0", "-6", "0", "-1" ], "h": [ "0", "1", "0", "1" ] }, "minimal_discriminant": "-25194240", "minimal_discriminant_factored": "-1 * 2^8 * 3^9 * 5", "conductor": "1440", "conductor_factored": "30 * 48 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 30 * 48 for the elliptic curves ['1', '0', '1', '-19', '26'] and ['0', '1', '0', '-24', '36'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "1440" }, "lmfdb": { "label": "1440.c.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B15%2C0%2C-6%2C0%2C-1%5D%2C%5B0%2C1%2C0%2C1%5D%5D", "snapshot_label": "1440.c.2" }, "torsion": { "invariants": [ 2, 24 ], "order": 48, "generators": [ { "order": 2, "mumford": [ "x^2 - 3", "0", "2" ] }, { "order": 24, "mumford": [ "x^2 - 3/2*x - 11/2", "-21/4*x - 39/4", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 96, "point_count_primes": [ 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 2, 24 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x : y : -z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "0", "1", "-19", "26" ], [ "0", "1", "0", "-24", "36" ] ], "conductors": [ "30", "48" ], "conductor": "1440", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 43 (source: \\cite{HLP2000})", "sources": [ "HLP2000" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "15", "0", "-6", "0", "-1" ], "h": [ "0", "1", "0", "1" ], "claimed_group": [ 2, 24 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "1440.c.2" }, "source": { "kind": "import", "name": "paper-t2-043", "file": "table2.txt", "line": 49, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:37:49 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.14000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.24.a.log", "conductor_log": "data/logs/2.24.a.conductor.log", "conductor_cputime_seconds": 0.2 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "2.24.b", "group": [ 2, 24 ], "group_key": "2.24", "group_bracket": "[2,24]", "order": 48, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "11816100", "-18141270", "-1242978", "27097", "6942", "588", "16" ], "h": [ "0", "1", "1" ], "f_str": "16*x^6 + 588*x^5 + 6942*x^4 + 27097*x^3 - 1242978*x^2 - 18141270*x + 11816100", "h_str": "x^2 + x", "g": [ "47264400", "-72565080", "-4971911", "108390", "27769", "2352", "64" ], "g_str": "64*x^6 + 2352*x^5 + 27769*x^4 + 108390*x^3 - 4971911*x^2 - 72565080*x + 47264400", "g_scale": "1", "discriminant": "-1383237283148003021143368021752539506459902767792128000000", "discriminant_digits": 58, "igusa_clebsch": [ "-5273373612856", "-141472441287627085941308", "-476523617997752906800303801038361528", "-1383237283148003021143368021752539506459902767792128000000" ], "g2_invariants": [ "124449390406778135846985128769718837701285198621227298524807/337704414831055425083830083435678590444312199168000000", "35660057551092970731326689947769708005457685974007159/2147591160657403751296233233082001618108416000000", "-7628709092989627043953296594865338513382522009/13657351194656871637771120987751841792000000" ], "rational_weierstrass_points": 3, "minimal_model": { "f": [ "75234825", "-3233367", "-1190247", "3043", "-2058", "12", "16" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "-337704414831055425083830083435678590444312199168000000", "minimal_discriminant_factored": "-1 * 2^54 * 3^15 * 5^6 * 7^5 * 11^3 * 13^5 * 17^7 * 24533", "conductor": "116250940866060", "conductor_factored": "2^2 * 3^2 * 5 * 7^2 * 11 * 13^2 * 17^2 * 24533", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous; computed prime by prime (conductor_by_prime.m), the exponent at 2 on the isomorphic model x -> (0 X + 1)/(1 X + 1)", "conductor_source": "magma" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B11816100%2C-18141270%2C-1242978%2C27097%2C6942%2C588%2C16%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 24 ], "order": 48, "generators": [ { "order": 2, "mumford": [ "x^2 + 3*x - 180", "0", "2" ] }, { "order": 24, "mumford": [ "x^2 + 203/8*x - 15", "-1350*x - 1140", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 48, "point_count_primes": [ 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131 ], "claimed": [ 2, 24 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 29, "chi": "T^4 - 2*T^3 - 14*T^2 - 58*T + 841", "primes_tried_below": 500, "admissible_primes": 3, "reducible_chi": 2 }, "q_simple": { "certified": true, "prime": 29, "chi": "T^4 - 2*T^3 - 14*T^2 - 58*T + 841" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 39, "reducible_chi": 14 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "A. V. Sutherland", "year": 2026, "historical": false }, "reference": "A. V. Sutherland, personal communication to F. Najman (November 2026)", "sources": [], "rm": false, "notes": "First geometrically simple Jacobian with torsion [2,24] in the census (the group was known only for Jacobians split over Q, 2.24.a). Found by A. V. Sutherland and sent to F. Najman as [[11816100,-18141270,-1242978,27097,6942,588,16],[0,1,1]]. End(J_Qbar) = Z: the Frobenius fields at 29 and 31 are quartic CM fields with no common quadratic subfield (data/knowledge/sources/sutherland_2_24_check.log).", "submitter": { "name": "Filip Najman (for A. V. Sutherland)", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "11816100", "-18141270", "-1242978", "27097", "6942", "588", "16" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 24 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "file", "path": "submissions/inbox/sutherland_2_24.json" }, "dates": { "submitted": "2026-11-10", "verified": "2026-11-10 12:02:24 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "0.650000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.24.b.log", "conductor_log": "data/logs/2.24.b.conductor.log", "conductor_cputime_seconds": 232.27 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.26.a", "group": [ 2, 26 ], "group_key": "2.26", "group_bracket": "[2,26]", "order": 52, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "0", "2", "3", "-12", "-3", "9" ], "h": [ "1", "0", "1" ], "f_str": "9*x^6 - 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3*x^5 - 42*x^4 + 34*x^3 + 4*x^2 - 3*x", "h_str": "x^2 + 1", "g": [ "1", "-12", "18", "136", "-167", "-12", "36" ], "g_str": "36*x^6 - 12*x^5 - 167*x^4 + 136*x^3 + 18*x^2 - 12*x + 1", "g_scale": "1", "discriminant": "79838576640000000", "discriminant_digits": 17, "igusa_clebsch": [ "156192", "619616064", "28029947822592", "79838576640000000" ], "g2_invariants": [ "34202591972030721/235000000", "1015178728390813/282000000", "23615146341289/338400000" ], "rational_weierstrass_points": 3, "minimal_model": { "f": [ "0", "-20", "61", "-27", "-7", "1" ], "h": [ "0", "0", "1", "1" ] }, "minimal_discriminant": "19491840000000", "minimal_discriminant_factored": "2^16 * 3^4 * 5^7 * 47", "conductor": "28200", "conductor_factored": "2^3 * 3 * 5^2 * 47", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "28200.e.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C-3%2C4%2C34%2C-42%2C-3%2C9%5D%2C%5B1%2C0%2C1%5D%5D", "snapshot_label": "28200.e.1" }, "torsion": { "invariants": [ 2, 28 ], "order": 56, "generators": [ { "order": 2, "mumford": [ "x^2 + 1/6*x - 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13*x^5 + 19*x^4 + 111*x^3 + 19*x^2 - 13*x + 1", "h_str": "x^2 + x", "g": [ "4", "-52", "77", "446", "77", "-52", "4" ], "g_str": "4*x^6 - 52*x^5 + 77*x^4 + 446*x^3 + 77*x^2 - 52*x + 4", "g_scale": "1", "discriminant": "243799621632000000000", "discriminant_digits": 21, "igusa_clebsch": [ "1202952", "26490176964", "9269921654726472", "243799621632000000000" ], "g2_invariants": [ "316362707828924813306843/244944000000000", "103565946608979088081/2721600000000", "1806512853604127/1440000000" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "1", "-13", "19", "111", "19", "-13", "1" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "59521392000000000", "minimal_discriminant_factored": "2^13 * 3^12 * 5^9 * 7", "conductor": "18900", "conductor_factored": "210 * 90 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 210 * 90 for the elliptic curves ['1', '0', '0', '-1070', '7812'] and ['1', '-1', '1', '-3002', '63929'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "18900" }, "lmfdb": { "label": "18900.e.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B1%2C-13%2C19%2C111%2C19%2C-13%2C1%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "18900.e.2" }, "torsion": { "invariants": [ 2, 48 ], "order": 96, "generators": [ { "order": 2, "mumford": [ "x^2 + 5/2*x + 1", "0", "2" ] }, { "order": 48, "mumford": [ "x^2 + 10/3*x - 2/3", "-415/9*x + 80/9", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 192, "point_count_primes": [ 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 2, 48 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 91, "reducible_chi": 91 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (z : y : x) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "0", "0", "-1070", "7812" ], [ "1", "-1", "1", "-3002", "63929" ] ], "conductors": [ "210", "90" ], "conductor": "18900", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 42, "reducible_chi": 42 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 45 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "1", "-13", "19", "111", "19", "-13", "1" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 48 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "18900.e.2" }, "source": { "kind": "import", "name": "paper-t2-045", "file": "table2.txt", "line": 51, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:37:53 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.16000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.48.a.log", "conductor_log": "data/logs/2.48.a.conductor.log", "conductor_cputime_seconds": 0.27 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "3.3.a", "group": [ 3, 3 ], "group_key": "3.3", "group_bracket": "[3,3]", "order": 9, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "3", "3", "7", "2", "4", "0", "1" ], "h": [ "1" ], "f_str": "x^6 + 4*x^4 + 2*x^3 + 7*x^2 + 3*x + 3", "h_str": "1", "g": [ "13", "12", "28", "8", "16", "0", "4" ], "g_str": "4*x^6 + 16*x^4 + 8*x^3 + 28*x^2 + 12*x + 13", "g_scale": "1", "discriminant": "-507617280000", "discriminant_digits": 12, "igusa_clebsch": [ "-19264", "1825600", "-11334003200", "-507617280000" ], "g2_invariants": [ "5060138806274048/7745625", "194243864639552/7745625", "113481738944/95625" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "3", "3", "7", "2", "4", "0", "1" ], "h": [ "1" ] }, "minimal_discriminant": "-123930000", "minimal_discriminant_factored": "-1 * 2^4 * 3^6 * 5^4 * 17", "conductor": "5100", "conductor_factored": "2^2 * 3 * 5^2 * 17", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "5100.a.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B3%2C3%2C7%2C2%2C4%2C0%2C1%5D%2C%5B1%5D%5D", "snapshot_label": "5100.a.1" }, "torsion": { "invariants": [ 3, 3 ], "order": 9, "generators": [ { "order": 3, "mumford": [ "x^2 + 1", "-2*x - 1", "2" ] }, { "order": 3, "mumford": [ "x^2 + x + 1/4", "5/2*x - 5/2", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 9, "point_count_primes": [ 7, 11, 13, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 3, 3 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 19, "chi": "T^4 - 4*T^3 + 15*T^2 - 76*T + 361", "primes_tried_below": 500, "admissible_primes": 4, "reducible_chi": 2 }, "q_simple": { "certified": true, "prime": 11, "chi": "T^4 + 4*T^2 + 121" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 42, "reducible_chi": 15 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 51 (source: \\cite{BFT2014})", "sources": [ "BFT2014" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "3", "3", "7", "2", "4", "0", "1" ], "h": [ "1" ], "claimed_group": [ 3, 3 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "5100.a.1" }, "source": { "kind": "import", "name": "paper-t1-051", "file": "table1.txt", "line": 57, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:49 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.370000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/3.3.a.log", "conductor_log": "data/logs/3.3.a.conductor.log", "conductor_cputime_seconds": 0.55 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "3.3.b", "group": [ 3, 3 ], "group_key": "3.3", "group_bracket": "[3,3]", "order": 9, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-2", "6", "-8", "6", "-8", "6", "-2" ], "h": [ "0" ], "f_str": "-2*x^6 + 6*x^5 - 8*x^4 + 6*x^3 - 8*x^2 + 6*x - 2", "h_str": "0", "g": [ "-8", "24", "-32", "24", "-32", "24", "-8" ], "g_str": "-8*x^6 + 24*x^5 - 32*x^4 + 24*x^3 - 32*x^2 + 24*x - 8", "g_scale": "1", "discriminant": "-117836722733056", "discriminant_digits": 15, "igusa_clebsch": [ "-5248", "3997696", "-1180696576", "-117836722733056" ], "g2_invariants": [ "115856201/27436", "-2687919/11552", "112627/1216" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "-2", "-6", "-8", "-6", "-8", "-6", "-2" ], "h": [] }, "minimal_discriminant": "-28768731136", "minimal_discriminant_factored": "-1 * 2^22 * 19^3", "conductor": "361", "conductor_factored": "19 * 19 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 19 * 19 for the elliptic curves ['0', '1', '1', '-9', '-15'] and ['0', '1', '1', '1', '0'] to whose product J is isogenous over Q", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "" }, "lmfdb": { "label": "361.a.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-2%2C6%2C-8%2C6%2C-8%2C6%2C-2%5D%2C%5B0%5D%5D", "snapshot_label": "361.a.1" }, "torsion": { "invariants": [ 3, 3 ], "order": 9, "generators": [ { "order": 3, "mumford": [ "x^2 - 3*x + 3", "-2", "2" ] }, { "order": 3, "mumford": [ "x^2 - x + 1", "2", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 9, "point_count_primes": [ 3, 5, 7, 11, 13, 17, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 3, 3 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 93, "reducible_chi": 93 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x - z : y : -z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "0", "1", "1", "-9", "-15" ], [ "0", "1", "1", "1", "0" ] ], "conductors": [ "19", "19" ], "conductor": "361", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 44, "reducible_chi": 44 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 46 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-2", "6", "-8", "6", "-8", "6", "-2" ], "h": [ "0" ], "claimed_group": [ 3, 3 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "361.a.1" }, "source": { "kind": "import", "name": "paper-t2-046", "file": "table2.txt", "line": 52, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:37:54 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.980000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/3.3.b.log", "conductor_log": "data/logs/3.3.b.conductor.log", "conductor_cputime_seconds": null }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "3.3.c", "group": [ 3, 3 ], "group_key": "3.3", "group_bracket": "[3,3]", "order": 9, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "196", "0", "0", "-13" ], "h": [ "0", "0", "0", "1" ], "f_str": "-13*x^3 + 196", "h_str": "x^3", "g": [ "784", "0", "0", "-52", "0", "0", "1" ], "g_str": "x^6 - 52*x^3 + 784", "g_scale": "1", "discriminant": "-36125252734943232", "discriminant_digits": 17, "igusa_clebsch": [ "-171936", "1682601984", "-73533803470848", "-36125252734943232" ], "g2_invariants": [ "1248318403996/2401", "9291226221/4802", "-23245787/9604" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "196", "0", "0", "-13" ], "h": [ "0", "0", "0", "1" ] }, "minimal_discriminant": "-8819641780992", "minimal_discriminant_factored": "-1 * 2^8 * 3^15 * 7^4", "conductor": "1764", "conductor_factored": "2^2 * 3^2 * 7^2", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "1764.d.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B196%2C0%2C0%2C-13%5D%2C%5B0%2C0%2C0%2C1%5D%5D", "snapshot_label": "1764.d.2" }, "torsion": { "invariants": [ 3, 3 ], "order": 9, "generators": [ { "order": 3, "mumford": [ "x", "-x^3 - 28", "2" ] }, { "order": 3, "mumford": [ "1", "x^3", "2" ] } ], "method": "generators", "generators_note": "", "generators_submitted": 2, "point_count_gcd": 9, "point_count_primes": [ 5, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 3, 3 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 71 }, "q_simple": { "certified": true, "prime": 43, "chi": "T^4 - 4*T^3 - 27*T^2 - 172*T + 1849" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "geometric_involution", "over": "Qbar", "detail": "the geometric automorphism group has order 12 (7 involutions), so the curve is bielliptic over Qbar" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 34 } }, "geometrically_isomorphic_to": [ "2.6.c" ], "discovery": { "by": "Booker–Sutherland (extended database)", "year": 2026, "historical": false }, "reference": "LMFDB extended genus 2 database (Booker–Sutherland), curve 1764.d.2 (table g2c_curves_new, label 1764.d2 on devmirror.lmfdb.xyz, read 2026-10-16)", "sources": [ "BookerSutherland" ], "rm": false, "notes": "First Q-simple, geometrically split example of [3,3] for the census (the gsplit cell is empty). Found by querying the LMFDB genus 2 data (devmirror.lmfdb.xyz, g2c_curves_new: is_simple_base = true, is_simple_geom = false, torsion_subgroup = [3,3]; geom_end_alg = M_2(Q), end_alg = Q; smallest conductor among the curves of this group that the census verifier certifies). LMFDB conductor 1764. Checked locally with the census's own pipeline/magma/verify_lib.m (Magma V2.30-1): J(Q)_tors = [3,3], class gsplit, Q-simplicity prime 43 (chi = T^4 - 4*T^3 - 27*T^2 - 172*T + 1849), splitness certificate: geometric_involution over Qbar (the geometric automorphism group has order 12 (7 involutions), so the curve is bielliptic over Qbar). Submitted by F. Najman; the curve is due to the database cited under Reference.", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "196", "0", "0", "-13" ], "h": [ "0", "0", "0", "1" ], "claimed_group": [ 3, 3 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "1764.d.2" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_3.3.json" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-17 06:12:17 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "1.22000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/3.3.c.log", "conductor_log": "data/logs/3.3.c.conductor.log", "conductor_cputime_seconds": 1.09 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "3.6.a", "group": [ 3, 6 ], "group_key": "3.6", "group_bracket": "[3,6]", "order": 18, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "21", "-2", "7", "-5", "0", "-1" ], "h": [ "1", "1", "1", "1" ], "f_str": "-x^5 - 5*x^3 + 7*x^2 - 2*x + 21", "h_str": "x^3 + x^2 + x + 1", "g": [ "85", "-6", "31", "-16", "3", "-2", "1" ], "g_str": "x^6 - 2*x^5 + 3*x^4 - 16*x^3 + 31*x^2 - 6*x + 85", "g_scale": "1", "discriminant": "-27869184000000", "discriminant_digits": 14, "igusa_clebsch": [ "-19872", "16708608", "-86482096128", "-27869184000000" ], "g2_invariants": [ "1520238471228/109375", "40922037621/218750", "307739931/62500" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "21", "-2", "7", "-5", "0", "-1" ], "h": [ "1", "1", "1", "1" ] }, "minimal_discriminant": "-6804000000", "minimal_discriminant_factored": "-1 * 2^8 * 3^5 * 5^6 * 7", "conductor": "3780", "conductor_factored": "2^2 * 3^3 * 5 * 7", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "3780.b.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B21%2C-2%2C7%2C-5%2C0%2C-1%5D%2C%5B1%2C1%2C1%2C1%5D%5D", "snapshot_label": "3780.b.1" }, "torsion": { "invariants": [ 3, 6 ], "order": 18, "generators": [ { "order": 3, "mumford": [ "x - 1/2", "-x^3 - 37/4", "2" ] }, { "order": 6, "mumford": [ "x - 3", "-x^3 + 7", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 18, "point_count_primes": [ 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 3, 6 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 13, "chi": "T^4 + 2*T^3 + 26*T + 169", "primes_tried_below": 500, "admissible_primes": 2, "reducible_chi": 1 }, "q_simple": { "certified": true, "prime": 13, "chi": "T^4 + 2*T^3 + 26*T + 169" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 42, "reducible_chi": 15 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 52 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "21", "-2", "7", "-5", "0", "-1" ], "h": [ "1", "1", "1", "1" ], "claimed_group": [ 3, 6 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "3780.b.1" }, "source": { "kind": "import", "name": "paper-t1-052", "file": "table1.txt", "line": 58, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:50 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.310000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/3.6.a.log", "conductor_log": "data/logs/3.6.a.conductor.log", "conductor_cputime_seconds": 0.51 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "3.6.b", "group": [ 3, 6 ], "group_key": "3.6", "group_bracket": "[3,6]", "order": 18, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "27", "-27", "56", "-13", "21", "8", "6" ], "h": [ "0", "1", "1" ], "f_str": "6*x^6 + 8*x^5 + 21*x^4 - 13*x^3 + 56*x^2 - 27*x + 27", "h_str": "x^2 + x", "g": [ "108", "-108", "225", "-50", "85", "32", "24" ], "g_str": "24*x^6 + 32*x^5 + 85*x^4 - 50*x^3 + 225*x^2 - 108*x + 108", "g_scale": "1", "discriminant": "-22220417174433988608", "discriminant_digits": 20, "igusa_clebsch": [ "-1051320", "21576373380", "-6528334812023160", "-22220417174433988608" ], "g2_invariants": [ "161294078381836186878125/22324718258136", "6932210817490695936875/33487077387204", "424388341784250875/62762119218" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "27", "-27", "56", "-13", "21", "8", "6" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "-5424906536727048", "minimal_discriminant_factored": "-1 * 2^3 * 3^24 * 7^4", "conductor": "196", "conductor_factored": "14 * 14 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 14 * 14 for the elliptic curves ['1', '0', '1', '-36', '-70'] and ['1', '0', '1', '-11', '12'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "196" }, "lmfdb": { "label": "196.a.4", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B27%2C-27%2C56%2C-13%2C21%2C8%2C6%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "196.a.4" }, "torsion": { "invariants": [ 3, 6 ], "order": 18, "generators": [ { "order": 3, "mumford": [ "x^2 - 6/23*x + 18/23", "-2517/529*x + 1548/529", "2" ] }, { "order": 6, "mumford": [ "x^2 + 4*x - 3/2", "-85*x + 39/2", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 36, "point_count_primes": [ 5, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 3, 6 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x - 3/2*z : 5/72*x^3 + 5/8*x^2*z + 15/8*x*z^2 + 27/8*y + 15/8*z^3 : -5/6*x - z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "0", "1", "-36", "-70" ], [ "1", "0", "1", "-11", "12" ] ], "conductors": [ "14", "14" ], "conductor": "196", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 47 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "27", "-27", "56", "-13", "21", "8", "6" ], "h": [ "0", "1", "1" ], "claimed_group": [ 3, 6 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "196.a.4" }, "source": { "kind": "import", "name": "paper-t2-047", "file": "table2.txt", "line": 53, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:37:55 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.31000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/3.6.b.log", "conductor_log": "data/logs/3.6.b.conductor.log", "conductor_cputime_seconds": 0.23 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "3.6.c", "group": [ 3, 6 ], "group_key": "3.6", "group_bracket": "[3,6]", "order": 18, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "19", "4", "-14", "11", "3", "-5", "2" ], "h": [ "1", "1", "1", "1" ], "f_str": "2*x^6 - 5*x^5 + 3*x^4 + 11*x^3 - 14*x^2 + 4*x + 19", "h_str": "x^3 + x^2 + x + 1", "g": [ "77", "18", "-53", "48", "15", "-18", "9" ], "g_str": "9*x^6 - 18*x^5 + 15*x^4 + 48*x^3 - 53*x^2 + 18*x + 77", "g_scale": "1", "discriminant": "1100753141760000", "discriminant_digits": 16, "igusa_clebsch": [ "-152736", "1507737600", "-57257725132800", "1100753141760000" ], "g2_invariants": [ "-10194147150028951/1080000", "12875176022987/960000", "4052580881/640000" ], "rational_weierstrass_points": 1, "minimal_model": { "f": [ "19", "4", "-14", "11", "3", "-5", "2" ], "h": [ "1", "1", "1", "1" ] }, "minimal_discriminant": "268738560000", "minimal_discriminant_factored": "2^16 * 3^8 * 5^4", "conductor": "2700", "conductor_factored": "2^2 * 3^3 * 5^2", "conductor_note": "taken from the LMFDB label 2700.b.1 (Magma did not finish the conductor computation within the time limit)", "conductor_source": "lmfdb label" }, "lmfdb": { "label": "2700.b.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B19%2C4%2C-14%2C11%2C3%2C-5%2C2%5D%2C%5B1%2C1%2C1%2C1%5D%5D", "snapshot_label": "2700.b.1" }, "torsion": { "invariants": [ 3, 6 ], "order": 18, "generators": [ { "order": 3, "mumford": [ "x - 1/3", "-3*x^3 - 79/9", "2" ] }, { "order": 6, "mumford": [ "x + 1", "-3*x^3 - 3", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "the submitted generators generate a subgroup of order 18, but the point-count bound is 36; TorsionSubgroup was run", "generators_submitted": 2, "point_count_gcd": 36, "point_count_primes": [ 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 3, 6 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 64 }, "q_simple": { "certified": true, "prime": 19, "chi": "T^4 + 5*T^3 + 6*T^2 + 95*T + 361" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "geometric_involution", "over": "Qbar", "detail": "the geometric automorphism group has order 12 (7 involutions), so the curve is bielliptic over Qbar" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 32 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": 2026, "historical": false }, "reference": "LMFDB extended genus 2 database (Booker–Sutherland), curve 2700.b.1 (table g2c_curves_new, label 2700.b1 on devmirror.lmfdb.xyz, read 2026-10-16)", "sources": [ "BookerSutherland" ], "rm": false, "notes": "First Q-simple, geometrically split example of [3,6] for the census (the gsplit cell is empty). Found by querying the LMFDB genus 2 data (devmirror.lmfdb.xyz, g2c_curves_new: is_simple_base = true, is_simple_geom = false, torsion_subgroup = [3,6]; geom_end_alg = M_2(Q), end_alg = Q; smallest conductor among the curves of this group that the census verifier certifies). LMFDB conductor 2700. Checked locally with the census's own pipeline/magma/verify_lib.m (Magma V2.30-1): J(Q)_tors = [3,6], class gsplit, Q-simplicity prime 19 (chi = T^4 + 5*T^3 + 6*T^2 + 95*T + 361), splitness certificate: geometric_involution over Qbar (the geometric automorphism group has order 12 (7 involutions), so the curve is bielliptic over Qbar). Submitted by F. Najman; the curve is due to the database cited under Reference.", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "19", "4", "-14", "11", "3", "-5", "2" ], "h": [ "1", "1", "1", "1" ], "claimed_group": [ 3, 6 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "2700.b.1" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_3.6.json" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-17 06:18:54 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "1.43000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/3.6.c.log", "conductor_log": "data/logs/3.6.c.conductor.log", "conductor_cputime_seconds": null }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "3.9.a", "group": [ 3, 9 ], "group_key": "3.9", "group_bracket": "[3,9]", "order": 27, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "326", "-75", "-230", "-113", "40", "75", "16" ], "h": [ "1", "0", "0", "1" ], "f_str": "16*x^6 + 75*x^5 + 40*x^4 - 113*x^3 - 230*x^2 - 75*x + 326", "h_str": "x^3 + 1", "g": [ "1305", "-300", "-920", "-450", "160", "300", "65" ], "g_str": "65*x^6 + 300*x^5 + 160*x^4 - 450*x^3 - 920*x^2 - 300*x + 1305", "g_scale": "1", "discriminant": "69185745000000000000000000000", "discriminant_digits": 29, "igusa_clebsch": [ "-20387800", "21189062500", "-32100838214609375000", "69185745000000000000000000000" ], "g2_invariants": [ "-11007833315061637294313699/1729643625000000000", "-57285703340477352129697/216205453125000000", "12949410971644987/1951171875000" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "265", "165", "-159", "152", "-92", "-23", "16" ], "h": [ "0", "1", "1", "1" ] }, "minimal_discriminant": "16891051025390625000000000", "minimal_discriminant_factored": "2^9 * 3^9 * 5^22 * 19 * 37", "conductor": "8436", "conductor_factored": "2^2 * 3 * 19 * 37", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "8436.a.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B326%2C-75%2C-230%2C-113%2C40%2C75%2C16%5D%2C%5B1%2C0%2C0%2C1%5D%5D", "snapshot_label": "8436.a.2" }, "torsion": { "invariants": [ 3, 9 ], "order": 27, "generators": [ { "order": 3, "mumford": [ "x^2 + 5/2*x + 3", "25/4*x - 75/2", "2" ] }, { "order": 9, "mumford": [ "x^2 + 5/4*x - 3/4", "-275/16*x - 375/16", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 27, "point_count_primes": [ 7, 11, 17, 23, 29, 31, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127 ], "claimed": [ 3, 9 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 17, "chi": "T^4 - 3*T^3 + 7*T^2 - 51*T + 289", "primes_tried_below": 500, "admissible_primes": 3, "reducible_chi": 2 }, "q_simple": { "certified": true, "prime": 17, "chi": "T^4 - 3*T^3 + 7*T^2 - 51*T + 289" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 40, "reducible_chi": 13 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 53 (source: \\cite{Leprevost1995})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "326", "-75", "-230", "-113", "40", "75", "16" ], "h": [ "1", "0", "0", "1" ], "claimed_group": [ 3, 9 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "8436.a.2" }, "source": { "kind": "import", "name": "paper-t1-053", "file": "table1.txt", "line": 59, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:51 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.350000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/3.9.a.log", "conductor_log": "data/logs/3.9.a.conductor.log", "conductor_cputime_seconds": 0.55 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "3.9.b", "group": [ 3, 9 ], "group_key": "3.9", "group_bracket": "[3,9]", "order": 27, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "0", "-2", "0", "5", "5", "3", "1" ], "h": [ "1", "1", "1" ], "f_str": "x^6 + 3*x^5 + 5*x^4 + 5*x^3 - 2*x", "h_str": "x^2 + x + 1", "g": [ "1", "-6", "3", "22", "21", "12", "4" ], "g_str": "4*x^6 + 12*x^5 + 21*x^4 + 22*x^3 + 3*x^2 - 6*x + 1", "g_scale": "1", "discriminant": "-573308928", "discriminant_digits": 9, "igusa_clebsch": [ "-1944", "453924", "-384120792", "-573308928" ], "g2_invariants": [ "387420489/64", "-3720087/16", "-132678" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "0", "1", "0", "-6", "5", "-3", "1" ], "h": [ "1", "1", "1" ] }, "minimal_discriminant": "-139968", "minimal_discriminant_factored": "-1 * 2^6 * 3^7", "conductor": "2916", "conductor_factored": "54 * 54 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 54 * 54 for the elliptic curves ['1', '-1', '1', '-14', '29'] and ['1', '-1', '0', '-3', '3'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "2916" }, "lmfdb": { "label": "2916.e.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C-2%2C0%2C5%2C5%2C3%2C1%5D%2C%5B1%2C1%2C1%5D%5D", "snapshot_label": "2916.e.2" }, "torsion": { "invariants": [ 3, 9 ], "order": 27, "generators": [ { "order": 3, "mumford": [ "x", "2*x^3 - 1", "2" ] }, { "order": 9, "mumford": [ "x^2 + 2*x + 1", "-3*x - 4", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 27, "point_count_primes": [ 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 3, 9 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 93, "reducible_chi": 93 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x + z : x^2*z + x*z^2 + y + z^3 : -z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "-1", "1", "-14", "29" ], [ "1", "-1", "0", "-3", "3" ] ], "conductors": [ "54", "54" ], "conductor": "2916", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 44, "reducible_chi": 44 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 48 (source: \\cite{HLP2000})", "sources": [ "HLP2000" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "-2", "0", "5", "5", "3", "1" ], "h": [ "1", "1", "1" ], "claimed_group": [ 3, 9 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "2916.e.2" }, "source": { "kind": "import", "name": "paper-t2-048", "file": "table2.txt", "line": 54, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:37:57 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.960000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/3.9.b.log", "conductor_log": "data/logs/3.9.b.conductor.log", "conductor_cputime_seconds": 0.58 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "3.9.c", "group": [ 3, 9 ], "group_key": "3.9", "group_bracket": "[3,9]", "order": 27, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "1", "-3", "-1", "7", "6", "4", "2" ], "h": [ "0", "1", "1", "1" ], "f_str": "2*x^6 + 4*x^5 + 6*x^4 + 7*x^3 - x^2 - 3*x + 1", "h_str": "x^3 + x^2 + x", "g": [ "4", "-12", "-3", "30", "27", "18", "9" ], "g_str": "9*x^6 + 18*x^5 + 27*x^4 + 30*x^3 - 3*x^2 - 12*x + 4", "g_scale": "1", "discriminant": "-975198486528", "discriminant_digits": 12, "igusa_clebsch": [ "-10584", "11573604", "-49327120152", "-975198486528" ], "g2_invariants": [ "1089547389/64", "-7411887/16", "-264453" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "2", "4", "6", "7", "-1", "-3", "1" ], "h": [ "1", "1", "1" ] }, "minimal_discriminant": "-238085568", "minimal_discriminant_factored": "-1 * 2^6 * 3^12 * 7", "conductor": "6804", "conductor_factored": "2^2 * 3^5 * 7", "conductor_note": "taken from the LMFDB label 6804.a.1 (Magma did not finish the conductor computation within the time limit)", "conductor_source": "lmfdb label" }, "lmfdb": { "label": "6804.a.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B1%2C-3%2C-1%2C7%2C6%2C4%2C2%5D%2C%5B0%2C1%2C1%2C1%5D%5D", "snapshot_label": "6804.a.1" }, "torsion": { "invariants": [ 3, 9 ], "order": 27, "generators": [ { "order": 3, "mumford": [ "x^2 + x", "x + 2", "2" ] }, { "order": 9, "mumford": [ "x^2 + x", "3*x + 2", "2" ] } ], "method": "generators", "generators_note": "", "generators_submitted": 2, "point_count_gcd": 27, "point_count_primes": [ 5, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 3, 9 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 56 }, "q_simple": { "certified": true, "prime": 17, "chi": "T^4 + 7*T^2 + 289" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "cover", "over": "X^2 + 3", "detail": "a nonconstant map of degree 3 from y^2 = g to the genus-1 curve v^2 = (-24*w)u^3 + (1/2*(27*w - 117))u^2 + (27*w + 9)u + (9) over X^2 + 3, (x,y) -> (p/q, y*hh/q^2) with p = x^2 + x, q = x^3 + (-w + 1)*x^2 + 1/6*(-5*w - 3)*x + 1/3*(w - 1), hh = x^3 + 1/2*(-w + 3)*x^2 - 2/3*w*x + 1/3*(-w - 1); degree-3 map, defined over Q(sqrt(-3)), from y^2 = g to a genus-1 curve; found by code/gsplit_cover3.m (torsion_jac, 2026-10-16)" }, "isogenous_product": null, "cover_submitted": true, "cover_accepted": true, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 25 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Leprévost 1995", "year": 1995, "historical": false }, "reference": "LMFDB extended genus 2 database (Booker–Sutherland), curve 6804.a.1 (table g2c_curves_new, label 6804.a1 on devmirror.lmfdb.xyz, read 2026-10-16); it is the curve C_3 of F. Leprévost, Jacobiennes de certaines courbes de genre 2: torsion et simplicité, J. Théor. Nombres Bordeaux 7 (1995), 283–306, p. 303 (the member t = 3 of the [3,9] family C_t), reprinted in Platonov–Zhgun–Petrunin 2013, p. 321 (credit corrected 2026-10-20)", "sources": [ "Leprevost1995" ], "rm": false, "notes": "First Q-simple, geometrically split example of [3,9] for the census (the gsplit cell is empty). Found by querying the LMFDB genus 2 data (devmirror.lmfdb.xyz, g2c_curves_new: is_simple_base = true, is_simple_geom = false); smallest-conductor curve of this torsion group there (LMFDB conductor 6804). The automatic Richelot certificates do not apply (over Q(sqrt(-3)) the working model factors as two cubics and has no Richelot isogeny), so a splitness certificate is supplied: an explicit degree-3 map to a genus-1 curve over Q(sqrt(-3)) (the field over which J splits; the cubic factors of g are the fibre over the point at infinity and the three remaining Weierstrass points). Checked locally with the census's own pipeline/magma/verify_lib.m (Magma V2.30-1). Submitted by F. Najman; the curve is Leprévost's C_3 (1995, see Reference), found in the database cited there, and the cover was computed for this submission.", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "1", "-3", "-1", "7", "6", "4", "2" ], "h": [ "0", "1", "1", "1" ], "claimed_group": [ 3, 9 ], "claimed_class": "split", "cover": { "field": "X^2+3", "cremona": "", "cubic": [ "-24*w", "1/2*(27*w-117)", "27*w+9", "9" ], "p": [ "0", "1", "1" ], "q": [ "1/3*(w-1)", "1/6*(-5*w-3)", "-w+1", "1" ], "hh": [ "1/3*(-w-1)", "-2/3*w", "1/2*(-w+3)", "1" ], "note": "degree-3 map, defined over Q(sqrt(-3)), from y^2 = g to a genus-1 curve; found by code/gsplit_cover3.m (torsion_jac, 2026-10-16)" }, "lmfdb_label_given": "6804.a.1" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_3.9.json" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-17 06:28:55 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "0.980000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/3.9.c.log", "conductor_log": "data/logs/3.9.c.conductor.log", "conductor_cputime_seconds": null }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "3.12.a", "group": [ 3, 12 ], "group_key": "3.12", "group_bracket": "[3,12]", "order": 36, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "-8", "12", "6", "-11", "-3", "3", "1" ], "h": [ "1", "1", "1" ], "f_str": "x^6 + 3*x^5 - 3*x^4 - 11*x^3 + 6*x^2 + 12*x - 8", "h_str": "x^2 + x + 1", "g": [ "-31", "50", "27", "-42", "-11", "12", "4" ], "g_str": "4*x^6 + 12*x^5 - 11*x^4 - 42*x^3 + 27*x^2 + 50*x - 31", "g_scale": "1", "discriminant": "13759414272000", "discriminant_digits": 14, "igusa_clebsch": [ "69096", "414756", "2387542824", "13759414272000" ], "g2_invariants": [ "197791818955790399/13824000", "11430375773281/19200", "2660653761/80" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "-8", "12", "6", "-11", "-3", "3", "1" ], "h": [ "1", "1", "1" ] }, "minimal_discriminant": "3359232000", "minimal_discriminant_factored": "2^12 * 3^8 * 5^3", "conductor": "2700", "conductor_factored": "90 * 30 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 90 * 30 for the elliptic curves ['1', '-1', '1', '-122', '1721'] and ['1', '0', '1', '1', '2'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "2700" }, "lmfdb": { "label": "2700.c.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-8%2C12%2C6%2C-11%2C-3%2C3%2C1%5D%2C%5B1%2C1%2C1%5D%5D", "snapshot_label": "2700.c.1" }, "torsion": { "invariants": [ 3, 12 ], "order": 36, "generators": [ { "order": 3, "mumford": [ "x^2 + x - 2", "-2*x - 1", "2" ] }, { "order": 12, "mumford": [ "x^2 + 1/3*x - 7/3", "-10/9*x - 32/9", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 144, "point_count_primes": [ 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 3, 12 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x + z : x^2*z + x*z^2 + y + z^3 : -z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "-1", "1", "-122", "1721" ], [ "1", "0", "1", "1", "2" ] ], "conductors": [ "90", "30" ], "conductor": "2700", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 49 (source: \\cite{HLP2000})", "sources": [ "HLP2000" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-8", "12", "6", "-11", "-3", "3", "1" ], "h": [ "1", "1", "1" ], "claimed_group": [ 3, 12 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "2700.c.1" }, "source": { "kind": "import", "name": "paper-t2-049", "file": "table2.txt", "line": 55, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:37:59 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.18000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/3.12.a.log", "conductor_log": "data/logs/3.12.a.conductor.log", "conductor_cputime_seconds": 0.27 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "3.12.b", "group": [ 3, 12 ], "group_key": "3.12", "group_bracket": "[3,12]", "order": 36, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "19644345", "10277505", "231383", "-7653746", "-1261612", "475068", "229392" ], "h": [ "0", "1", "1" ], "f_str": "229392*x^6 + 475068*x^5 - 1261612*x^4 - 7653746*x^3 + 231383*x^2 + 10277505*x + 19644345", "h_str": "x^2 + x", "g": [ "78577380", "41110020", "925533", "-30614982", "-5046447", "1900272", "917568" ], "g_str": "917568*x^6 + 1900272*x^5 - 5046447*x^4 - 30614982*x^3 + 925533*x^2 + 41110020*x + 78577380", "g_scale": "1", "discriminant": "332044463560069793610209703254495927995614958486684933354547835637334016000", "discriminant_digits": 75, "igusa_clebsch": [ "-8480819472962040", "23262720633981746097751718090244", "-46660782332481755543015380104303346690374147576", "332044463560069793610209703254495927995614958486684933354547835637334016000" ], "g2_invariants": [ "-4381385463391418683164385967030389024203833744475/265282498890203682024091290083429809884495872", "11908852197936782766234702306598590163615837629/4145039045159432531626426407553590779445248", "30948504982057486410937110471854831614431849/323831175403080666533314563090124279644160" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "19644345", "10277505", "231383", "-7653746", "-1261612", "475068", "229392" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "81065542861345164455617603333617169920804433224288313807262655184896000", "minimal_discriminant_factored": "2^45 * 3^14 * 5^3 * 7^4 * 17 * 47^3 * 59^22", "conductor": "951369300", "conductor_factored": "2^2 * 3^5 * 5^2 * 7^2 * 17 * 47", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B19644345%2C10277505%2C231383%2C-7653746%2C-1261612%2C475068%2C229392%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 3, 12 ], "order": 36, "generators": [ { "order": 3, "mumford": [ "x^2 + 291/11*x - 126/11", "-78907308/121*x + 34796076/121", "2" ] }, { "order": 12, "mumford": [ "x^2 - 65/46*x - 189/46", "3164229/1058*x - 5336373/1058", "2" ] } ], "method": "generators", "generators_note": "", "generators_submitted": 2, "point_count_gcd": 36, "point_count_primes": [ 11, 13, 19, 23, 29, 31, 37, 41, 43, 53, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131 ], "claimed": [ 3, 12 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 37, "chi": "T^4 - T^3 - 37*T + 1369", "primes_tried_below": 500, "admissible_primes": 7, "reducible_chi": 6 }, "q_simple": { "certified": true, "prime": 37, "chi": "T^4 - T^3 - 37*T + 1369" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 39, "reducible_chi": 21 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "F. Najman", "year": 2026, "historical": false }, "reference": "new", "sources": [], "rm": false, "notes": "First example of [3,12] = Z/3 x Z/12 for a geometrically simple genus 2 Jacobian over Q (previously known only for Jacobians split over Q).\n\nOutput of pipeline/magma/prepare_submission.m (Magma V2.29-4): J(Q)_tors = [3,12]; gcd of #J(F_p) over 25 good primes = 36 (sharp); strict prime 37, chi = x^4 - x^3 - 37*x + 1369. Independent PARI/GP check (hyperellcharpoly): gcd 36 over 71 good odd p < 400, 16 strict primes below 200. Sage/Lombardo: End(J_Qbar) = Z. #Aut(C) = 2. Conductor: the odd part is 3^5*5^2*7^2*17*47 (PARI genus2red); the exponent at 2 is not computed.\n\nHow it was found: by a sieve over the order-12 chart A(12), written in coordinates (A,z,B) = (p^2, z, p*r). The sieve kept points with Frobenius polynomial = (x-1)^2 (x-p)^2 mod 3 and 36 | #J(F_p) at good primes. The curve is at (A,z,B) = (-2/3, 5, -10). On the line (A,B) = (-2/3,-10), the extra rational 3-torsion point exists exactly over the rational points of the rank-1 elliptic curve w^2 = (3z^2+25)(3z^2+121) (conductor 990). This gives infinitely many geometrically simple Jacobians with torsion exactly [3,12] (a computational proof, independently re-verified). Six further non-isomorphic simple [3,12] curves were found.", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "19644345", "10277505", "231383", "-7653746", "-1261612", "475068", "229392" ], "h": [ "0", "1", "1" ], "claimed_group": [ 3, 12 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "issue", "number": 2, "url": "https://github.com/Genus-2-torsion/Genus-2-torsion.github.io/issues/2" }, "dates": { "submitted": "2026-10-12", "verified": "2026-10-12 18:02:46 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.270000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/3.12.b.log", "conductor_log": "data/logs/3.12.b.conductor.log", "conductor_cputime_seconds": 218.13 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "3.24.a", "group": [ 3, 24 ], "group_key": "3.24", "group_bracket": "[3,24]", "order": 72, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "4", "6", "-4", "-6", "5", "-3", "1" ], "h": [ "0", "1", "1" ], "f_str": "x^6 - 3*x^5 + 5*x^4 - 6*x^3 - 4*x^2 + 6*x + 4", "h_str": "x^2 + x", "g": [ "16", "24", "-15", "-22", "21", "-12", "4" ], "g_str": "4*x^6 - 12*x^5 + 21*x^4 - 22*x^3 - 15*x^2 + 24*x + 16", "g_scale": "1", "discriminant": "-110075314176000", "discriminant_digits": 15, "igusa_clebsch": [ "-18936", "16915716", "-115543469304", "-110075314176000" ], "g2_invariants": [ "11324557777887/4096000", "2892440073/102400", "-28843473/2560" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "4", "-6", "-4", "5", "5", "3", "1" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "-26873856000", "minimal_discriminant_factored": "-1 * 2^15 * 3^8 * 5^3", "conductor": "8100", "conductor_factored": "90 * 90 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 90 * 90 for the elliptic curves ['1', '-1', '1', '-122', '1721'] and ['1', '-1', '1', '-8', '11'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "8100" }, "lmfdb": { "label": "8100.e.4", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B4%2C6%2C-4%2C-6%2C5%2C-3%2C1%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "8100.e.4" }, "torsion": { "invariants": [ 3, 24 ], "order": 72, "generators": [ { "order": 3, "mumford": [ "x", "-2*x^3 - 4", "2" ] }, { "order": 24, "mumford": [ "x^2 - 2*x", "-5*x + 4", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 72, "point_count_primes": [ 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 3, 24 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x - z : x^2*z - x*z^2 + y + z^3 : -z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "-1", "1", "-122", "1721" ], [ "1", "-1", "1", "-8", "11" ] ], "conductors": [ "90", "90" ], "conductor": "8100", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 50 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "4", "6", "-4", "-6", "5", "-3", "1" ], "h": [ "0", "1", "1" ], "claimed_group": [ 3, 24 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "8100.e.4" }, "source": { "kind": "import", "name": "paper-t2-050", "file": "table2.txt", "line": 56, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:38:00 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.18000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/3.24.a.log", "conductor_log": "data/logs/3.24.a.conductor.log", "conductor_cputime_seconds": 0.3 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "4.4.a", "group": [ 4, 4 ], "group_key": "4.4", "group_bracket": "[4,4]", "order": 16, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "20493", "6534", "10832", "2601", "1823", "242", "99" ], "h": [ "0", "1", "1" ], "f_str": "99*x^6 + 242*x^5 + 1823*x^4 + 2601*x^3 + 10832*x^2 + 6534*x + 20493", "h_str": "x^2 + x", "g": [ "81972", "26136", "43329", "10406", "7293", "968", "396" ], "g_str": "396*x^6 + 968*x^5 + 7293*x^4 + 10406*x^3 + 43329*x^2 + 26136*x + 81972", "g_scale": "1", "discriminant": "-136782329214257138140529311304640000000", "discriminant_digits": 39, "igusa_clebsch": [ "-11184898296", "49692525742203300", "-174996613247544814426608600", "-136782329214257138140529311304640000000" ], "g2_invariants": [ "847573045761591729112618790232749/5298304593169069453125", "105273291169035906326982565036303/15894913779507208359375", "52112636136936741387633771539939/143054224015564875234375" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "20493", "6534", "10832", "2601", "1823", "242", "99" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "-33394123343324496616340163892734375", "minimal_discriminant_factored": "-1 * 3^7 * 5^7 * 7^4 * 11^22", "conductor": "1575", "conductor_factored": "3^2 * 5^2 * 7", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "1575.c.14", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B20493%2C6534%2C10832%2C2601%2C1823%2C242%2C99%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "1575.c.14" }, "torsion": { "invariants": [ 4, 4 ], "order": 16, "generators": [ { "order": 4, "mumford": [ "x^2 + 55/2*x + 63/2", "-13915*x - 16335", "2" ] }, { "order": 4, "mumford": [ "x^2 + 11/9*x + 4", "968/27*x - 242/3", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 16, "point_count_primes": [ 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113 ], "claimed": [ 4, 4 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 31, "chi": "T^4 + 4*T^3 - 2*T^2 + 124*T + 961", "primes_tried_below": 500, "admissible_primes": 6, "reducible_chi": 4 }, "q_simple": { "certified": true, "prime": 29, "chi": "T^4 - 10*T^2 + 841" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 41, "reducible_chi": 15 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 54 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "20493", "6534", "10832", "2601", "1823", "242", "99" ], "h": [ "0", "1", "1" ], "claimed_group": [ 4, 4 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "1575.c.14" }, "source": { "kind": "import", "name": "paper-t1-054", "file": "table1.txt", "line": 60, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:52 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.420000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/4.4.a.log", "conductor_log": "data/logs/4.4.a.conductor.log", "conductor_cputime_seconds": 0.24 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "4.4.b", "group": [ 4, 4 ], "group_key": "4.4", "group_bracket": "[4,4]", "order": 16, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "448", "0", "-7", "0", "-2", "0", "7" ], "h": [ "0", "0", "1" ], "f_str": "7*x^6 - 2*x^4 - 7*x^2 + 448", "h_str": "x^2", "g": [ "1792", "0", "-28", "0", "-7", "0", "28" ], "g_str": "28*x^6 - 7*x^4 - 28*x^2 + 1792", "g_scale": "1", "discriminant": "-14759080257020238995285606400", "discriminant_digits": 29, "igusa_clebsch": [ "-12045376", "4073546741824", "-15133595718874719744", "-14759080257020238995285606400" ], "g2_invariants": [ "6688234978541721608/3114289620225", "613965885430366081/12457158480900", "-5442426347776/13841287201" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "448", "0", "-7", "0", "-2", "0", "7" ], "h": [ "0", "0", "1" ] }, "minimal_discriminant": "-3603291078374081785958400", "minimal_discriminant_factored": "-1 * 2^12 * 3^2 * 5^2 * 7^22", "conductor": "225", "conductor_factored": "15 * 15 (product of the elliptic conductors)", "conductor_note": "Magma's genus-2 Conductor gave 230400 (Ogg's formula at 2, no correctness guarantee); the rigorous value is N(E1) N(E2) = 15 * 15 for the elliptic curves ['1', '1', '1', '-5', '2'] and ['1', '1', '1', '-5', '2'] to whose product J is isogenous over Q", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "230400" }, "lmfdb": { "label": "225.a.6", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B448%2C0%2C-7%2C0%2C-2%2C0%2C7%5D%2C%5B0%2C0%2C1%5D%5D", "snapshot_label": "225.a.6" }, "torsion": { "invariants": [ 4, 4 ], "order": 16, "generators": [ { "order": 4, "mumford": [ "x^2 + 7*x + 4", "245*x + 196", "2" ] }, { "order": 4, "mumford": [ "x^2 - 3", "49", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 64, "point_count_primes": [ 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 4, 4 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 91, "reducible_chi": 91 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (z : 1/16*x^2*z - 1/8*x*z^2 + 1/8*y : 1/4*x) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "1", "1", "-5", "2" ], [ "1", "1", "1", "-5", "2" ] ], "conductors": [ "15", "15" ], "conductor": "225", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 42, "reducible_chi": 42 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 51 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "448", "0", "-7", "0", "-2", "0", "7" ], "h": [ "0", "0", "1" ], "claimed_group": [ 4, 4 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "225.a.6" }, "source": { "kind": "import", "name": "paper-t2-051", "file": "table2.txt", "line": 57, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:38:01 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.01000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/4.4.b.log", "conductor_log": "data/logs/4.4.b.conductor.log", "conductor_cputime_seconds": 0.16 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "4.4.c", "group": [ 4, 4 ], "group_key": "4.4", "group_bracket": "[4,4]", "order": 16, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "1", "10", "-20", "-40", "735", "-1260", "-540" ], "h": [ "1" ], "f_str": "-540*x^6 - 1260*x^5 + 735*x^4 - 40*x^3 - 20*x^2 + 10*x + 1", "h_str": "1", "g": [ "5", "40", "-80", "-160", "2940", "-5040", "-2160" ], "g_str": "-2160*x^6 - 5040*x^5 + 2940*x^4 - 160*x^3 - 80*x^2 + 40*x + 5", "g_scale": "1", "discriminant": "-99202320000000000000000000000", "discriminant_digits": 29, "igusa_clebsch": [ "-1555200", "-6660000000000", "-17635572000000000000", "-99202320000000000000000000000" ], "g2_invariants": [ "19591041024/1708984375", "36780293376/1708984375", "-274772451104/5126953125" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "-540", "-1260", "735", "-40", "-20", "10", "1" ], "h": [ "0", "0", "0", "1" ] }, "minimal_discriminant": "-24219316406250000000000000", "minimal_discriminant_factored": "-1 * 2^13 * 3^11 * 5^22 * 7", "conductor": "48384", "conductor_factored": "2^8 * 3^3 * 7", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "48384.j.5", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B1%2C10%2C-20%2C-40%2C735%2C-1260%2C-540%5D%2C%5B1%5D%5D", "snapshot_label": "48384.j.5" }, "torsion": { "invariants": [ 4, 4 ], "order": 16, "generators": [ { "order": 4, "mumford": [ "x^2 - 1/8*x - 1/8", "-175/16*x + 25/16", "2" ] }, { "order": 4, "mumford": [ "x^2 - 11/48*x - 1/48", "925/64*x - 25/64", "2" ] } ], "method": "generators", "generators_note": "", "generators_submitted": 2, "point_count_gcd": 16, "point_count_primes": [ 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 4, 4 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 91, "reducible_chi": 55 }, "q_simple": { "certified": true, "prime": 17, "chi": "T^4 + 14*T^2 + 289" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "richelot", "over": "Q", "detail": "a chain of two (2,2)-isogenies over Q reaches a Weil restriction of an elliptic curve E over Q(sqrt(-3)) (the surface is E x E^sigma)" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 42, "reducible_chi": 26 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": 2026, "historical": false }, "reference": "LMFDB extended genus 2 database (Booker–Sutherland), curve 48384.j.5 (table g2c_curves_new, label 48384.j5 on devmirror.lmfdb.xyz, read 2026-10-16)", "sources": [ "BookerSutherland" ], "rm": false, "notes": "First Q-simple, geometrically split example of [4,4] for the census (the gsplit cell is empty). Found by querying the LMFDB genus 2 data (devmirror.lmfdb.xyz, g2c_curves_new: is_simple_base = true, is_simple_geom = false, torsion_subgroup = [4,4]; geom_end_alg = Q x Q, end_alg = Q; smallest conductor among the curves of this group that the census verifier certifies). LMFDB conductor 48384. Checked locally with the census's own pipeline/magma/verify_lib.m (Magma V2.30-1): J(Q)_tors = [4,4], class gsplit, Q-simplicity prime 17 (chi = T^4 + 14*T^2 + 289), splitness certificate: richelot over Q (a chain of two (2,2)-isogenies over Q reaches a Weil restriction of an elliptic curve E over Q(sqrt(-3)) (the surface is E x E^sigma)). Submitted by F. Najman; the curve is due to the database cited under Reference. Rigorous conductor: J is (2,2)-isogenous over Q to Res_(K/Q) E with K the quadratic field of discriminant -3, so N(J) = |d_K|^2 * Norm(f_E) = 9 * 5376 = 48384 (computed with Magma; conductor is an isogeny invariant).", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "1", "10", "-20", "-40", "735", "-1260", "-540" ], "h": [ "1" ], "claimed_group": [ 4, 4 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "48384.j.5" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_4.4.json" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-17 06:28:57 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "0.770000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/4.4.c.log", "conductor_log": "data/logs/4.4.c.conductor.log", "conductor_cputime_seconds": 0.22 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "4.8.a", "group": [ 4, 8 ], "group_key": "4.8", "group_bracket": "[4,8]", "order": 32, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "164", "682", "450", "-435", "272", "-81" ], "h": [ "0", "1", "1" ], "f_str": "-81*x^5 + 272*x^4 - 435*x^3 + 450*x^2 + 682*x + 164", "h_str": "x^2 + x", "g": [ "656", "2728", "1801", "-1738", "1089", "-324" ], "g_str": "-324*x^5 + 1089*x^4 - 1738*x^3 + 1801*x^2 + 2728*x + 656", "g_scale": "1", "discriminant": "-30692353397852160000000000", "discriminant_digits": 26, "igusa_clebsch": [ "-48611640", "-99142856919900", "633597558673979529000", "-3221960490292928348160000000000" ], "g2_invariants": [ "4543630171735017667068457/431431456550400000", "1328890197975492557125963/1812012117511680000", "-2707867945940262321821/465945973074432000" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "164", "-682", "450", "434", "272", "81" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "-786611447825421960000000000", "minimal_discriminant_factored": "-1 * 2^12 * 3^24 * 5^10 * 7^4 * 29", "conductor": "913500", "conductor_factored": "2^2 * 3^2 * 5^3 * 7 * 29", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "913500.cf.3", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B164%2C682%2C450%2C-435%2C272%2C-81%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "913500.cf.3" }, "torsion": { "invariants": [ 4, 8 ], "order": 32, "generators": [ { "order": 4, "mumford": [ "x^2 + 19/36*x - 1/36", "-65*x - 25", "2" ] }, { "order": 8, "mumford": [ "x^2 + 73/9*x - 100/9", "548*x - 584", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 32, "point_count_primes": [ 11, 13, 17, 19, 23, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113 ], "claimed": [ 4, 8 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 53, "chi": "T^4 - 4*T^3 + 62*T^2 - 212*T + 2809", "primes_tried_below": 500, "admissible_primes": 11, "reducible_chi": 9 }, "q_simple": { "certified": true, "prime": 19, "chi": "T^4 - 10*T^2 + 361" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 41, "reducible_chi": 23 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 55 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "164", "682", "450", "-435", "272", "-81" ], "h": [ "0", "1", "1" ], "claimed_group": [ 4, 8 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "913500.cf.3" }, "source": { "kind": "import", "name": "paper-t1-055", "file": "table1.txt", "line": 61, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:53 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.400000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/4.8.a.log", "conductor_log": "data/logs/4.8.a.conductor.log", "conductor_cputime_seconds": 0.35 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "4.8.b", "group": [ 4, 8 ], "group_key": "4.8", "group_bracket": "[4,8]", "order": 32, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "0", "-15", "-5", "-5" ], "h": [ "0", "1", "0", "1" ], "f_str": "-5*x^3 - 5*x^2 - 15*x", "h_str": "x^3 + x", "g": [ "0", "-60", "-19", "-20", "2", "0", "1" ], "g_str": "x^6 + 2*x^4 - 20*x^3 - 19*x^2 - 60*x", "g_scale": "1", "discriminant": "1415577600000", "discriminant_digits": 13, "igusa_clebsch": [ "3008", "5882944", "11645495808", "1415577600000" ], "g2_invariants": [ "1834760056/84375", "-575075597/67500", "-700253/125" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "0", "-15", "-5", "-5" ], "h": [ "0", "1", "0", "1" ] }, "minimal_discriminant": "345600000", "minimal_discriminant_factored": "2^12 * 3^3 * 5^5", "conductor": "225", "conductor_factored": "15 * 15 (product of the elliptic conductors)", "conductor_note": "Magma's genus-2 Conductor gave 230400 (Ogg's formula at 2, no correctness guarantee); the rigorous value is N(E1) N(E2) = 15 * 15 for the elliptic curves ['1', '1', '1', '-5', '2'] and ['1', '1', '1', '-10', '-10'] to whose product J is isogenous over Q", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "230400" }, "lmfdb": { "label": "225.a.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C-15%2C-5%2C-5%5D%2C%5B0%2C1%2C0%2C1%5D%5D", "snapshot_label": "225.a.2" }, "torsion": { "invariants": [ 4, 8 ], "order": 32, "generators": [ { "order": 4, "mumford": [ "x + 1", "-x^3 + 7", "2" ] }, { "order": 8, "mumford": [ "x^2 + x", "8*x", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 64, "point_count_primes": [ 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 4, 8 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x - 3*z : 3*x^3 + 5*x^2*z - 7*x*z^2 + 8*y + 15*z^3 : -x - z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "1", "1", "-5", "2" ], [ "1", "1", "1", "-10", "-10" ] ], "conductors": [ "15", "15" ], "conductor": "225", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 52 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "-15", "-5", "-5" ], "h": [ "0", "1", "0", "1" ], "claimed_group": [ 4, 8 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "225.a.2" }, "source": { "kind": "import", "name": "paper-t2-052", "file": "table2.txt", "line": 58, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:38:03 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.970000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/4.8.b.log", "conductor_log": "data/logs/4.8.b.conductor.log", "conductor_cputime_seconds": 0.24 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "4.12.a", "group": [ 4, 12 ], "group_key": "4.12", "group_bracket": "[4,12]", "order": 48, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "9", "-27", "53", "-62", "53", "-27", "9" ], "h": [ "0", "1", "1" ], "f_str": "9*x^6 - 27*x^5 + 53*x^4 - 62*x^3 + 53*x^2 - 27*x + 9", "h_str": "x^2 + x", "g": [ "36", "-108", "213", "-246", "213", "-108", "36" ], "g_str": "36*x^6 - 108*x^5 + 213*x^4 - 246*x^3 + 213*x^2 - 108*x + 36", "g_scale": "1", "discriminant": "-3343537668096000", "discriminant_digits": 16, "igusa_clebsch": [ "-207288", "3732804", "-64463656248", "-3343537668096000" ], "g2_invariants": [ "197791818955790399/13824000", "11430375773281/19200", "2660653761/80" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "9", "27", "53", "61", "53", "27", "9" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "-816293376000", "minimal_discriminant_factored": "-1 * 2^12 * 3^13 * 5^3", "conductor": "8100", "conductor_factored": "90 * 90 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 90 * 90 for the elliptic curves ['1', '-1', '1', '13', '-61'] and ['1', '-1', '1', '-122', '1721'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "8100" }, "lmfdb": { "label": "8100.j.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B9%2C-27%2C53%2C-62%2C53%2C-27%2C9%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "8100.j.1" }, "torsion": { "invariants": [ 4, 12 ], "order": 48, "generators": [ { "order": 4, "mumford": [ "x^2 - x + 3/2", "-3*x + 3/2", "2" ] }, { "order": 12, "mumford": [ "x^2 - 5/3*x + 2", "2/3*x + 4", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 144, "point_count_primes": [ 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 4, 12 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x : -x^3 + 2*x^2*z + y : x - z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "-1", "1", "13", "-61" ], [ "1", "-1", "1", "-122", "1721" ] ], "conductors": [ "90", "90" ], "conductor": "8100", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [ "3.12.a" ], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 53 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "9", "-27", "53", "-62", "53", "-27", "9" ], "h": [ "0", "1", "1" ], "claimed_group": [ 4, 12 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "8100.j.1" }, "source": { "kind": "import", "name": "paper-t2-053", "file": "table2.txt", "line": 59, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:38:04 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.21000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/4.12.a.log", "conductor_log": "data/logs/4.12.a.conductor.log", "conductor_cputime_seconds": 0.28 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "4.12.b", "group": [ 4, 12 ], "group_key": "4.12", "group_bracket": "[4,12]", "order": 48, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "9", "165", "579", "-1653", "570", "45" ], "h": [ "0", "1", "1" ], "f_str": "45*x^5 + 570*x^4 - 1653*x^3 + 579*x^2 + 165*x + 9", "h_str": "x^2 + x", "g": [ "36", "660", "2317", "-6610", "2281", "180" ], "g_str": "180*x^5 + 2281*x^4 - 6610*x^3 + 2317*x^2 + 660*x + 36", "g_scale": "1", "discriminant": "9662005946592000000000", "discriminant_digits": 22, "igusa_clebsch": [ "182343368", "687908192950564", "35927588984325443563688", "313048992669580800000000000" ], "g2_invariants": [ "6151767682137268581810194611794108601/76427976725971875000000", "28225879153202209451898660474157/12580736909625000000", "439406929571516673890901587/6212709585000000" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "9", "165", "579", "-1653", "570", "45" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "76427976725971875000000", "minimal_discriminant_factored": "2^6 * 3^12 * 5^11 * 17^4 * 19 * 29", "conductor": "8430300", "conductor_factored": "2^2 * 3^2 * 5^2 * 17 * 19 * 29", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B9%2C165%2C579%2C-1653%2C570%2C45%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 4, 12 ], "order": 48, "generators": [ { "order": 4, "mumford": [ "x^2 - 25/12*x - 1/4", "-185/6*x - 5/2", "2" ] }, { "order": 12, "mumford": [ "x^2 + 41/3*x + 4/3", "-754/3*x - 74/3", "2" ] } ], "method": "generators", "generators_note": "", "generators_submitted": 2, "point_count_gcd": 48, "point_count_primes": [ 7, 11, 13, 23, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127 ], "claimed": [ 4, 12 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 31, "chi": "T^4 + 8*T^3 + 30*T^2 + 248*T + 961", "primes_tried_below": 500, "admissible_primes": 5, "reducible_chi": 4 }, "q_simple": { "certified": true, "prime": 31, "chi": "T^4 + 8*T^3 + 30*T^2 + 248*T + 961" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 40, "reducible_chi": 24 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "F. Najman", "year": 2026, "historical": false }, "reference": "new", "sources": [], "rm": false, "notes": "First example of [4,12] = Z/4 x Z/12 for a geometrically simple genus 2 Jacobian over Q (previously known only for Jacobians split over Q).\n\nOutput of pipeline/magma/prepare_submission.m (Magma V2.29-4): J(Q)_tors = [4,12]; gcd of #J(F_p) over 25 good primes = 48 (sharp); strict prime 31, chi = x^4 + 8*x^3 + 30*x^2 + 248*x + 961; conductor 8430300 = 2^2*3^2*5^2*17*19*29. Independent PARI/GP check (hyperellcharpoly): gcd 48 over 103 good odd p < 600, 10 strict primes below 200. Sage/Lombardo: End(J_Qbar) = Z. #Aut(C) = 2.\n\nHow it was found: by a search on the order-12 chart A(12), written in coordinates (A,z,B) = (p^2, z, p*r), restricted to the plane A = 1/4, where all three 2-torsion points are rational. The curve is at (A,z,B) = (1/4, 10/3, 127/36). Eleven further pairwise non-isomorphic simple [4,12] curves were found on the same plane; no infinite family is known.", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "9", "165", "579", "-1653", "570", "45" ], "h": [ "0", "1", "1" ], "claimed_group": [ 4, 12 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "issue", "number": 3, "url": "https://github.com/Genus-2-torsion/Genus-2-torsion.github.io/issues/3" }, "dates": { "submitted": "2026-10-13", "verified": "2026-10-13 08:22:34 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.200000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/4.12.b.log", "conductor_log": "data/logs/4.12.b.conductor.log", "conductor_cputime_seconds": 0.23 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "4.16.a", "group": [ 4, 16 ], "group_key": "4.16", "group_bracket": "[4,16]", "order": 64, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "252", "0", "-24", "0", "-4" ], "h": [ "0", "1", "0", "1" ], "f_str": "-4*x^4 - 24*x^2 + 252", "h_str": "x^3 + x", "g": [ "1008", "0", "-95", "0", "-14", "0", "1" ], "g_str": "x^6 - 14*x^4 - 95*x^2 + 1008", "g_scale": "1", "discriminant": "-10835538739200000000", "discriminant_digits": 20, "igusa_clebsch": [ "-263200", "803497024", "-93034372985344", "-10835538739200000000" ], "g2_invariants": [ "13766434045175/944784", "648822423599/1312200", "-483620788/455625" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "252", "0", "-24", "0", "-4" ], "h": [ "0", "1", "0", "1" ] }, "minimal_discriminant": "-2645395200000000", "minimal_discriminant_factored": "-1 * 2^14 * 3^10 * 5^8 * 7", "conductor": "50400", "conductor_factored": "210 * 240 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 210 * 240 for the elliptic curves ['1', '0', '0', '-1070', '7812'] and ['0', '1', '0', '-160', '308'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "50400" }, "lmfdb": { "label": "50400.dj.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B252%2C0%2C-24%2C0%2C-4%5D%2C%5B0%2C1%2C0%2C1%5D%5D", "snapshot_label": "50400.dj.1" }, "torsion": { "invariants": [ 4, 16 ], "order": 64, "generators": [ { "order": 4, "mumford": [ "x^2 - 5*x + 9", "-15*x + 75", "2" ] }, { "order": 16, "mumford": [ "x^2 - 2*x - 9", "6*x - 6", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 128, "point_count_primes": [ 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 4, 16 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 91, "reducible_chi": 91 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x : y : -z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "0", "0", "-1070", "7812" ], [ "0", "1", "0", "-160", "308" ] ], "conductors": [ "210", "240" ], "conductor": "50400", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 42, "reducible_chi": 42 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 54 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "252", "0", "-24", "0", "-4" ], "h": [ "0", "1", "0", "1" ], "claimed_group": [ 4, 16 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "50400.dj.1" }, "source": { "kind": "import", "name": "paper-t2-054", "file": "table2.txt", "line": 60, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:38:06 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.20000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/4.16.a.log", "conductor_log": "data/logs/4.16.a.conductor.log", "conductor_cputime_seconds": 0.26 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "5.5.a", "group": [ 5, 5 ], "group_key": "5.5", "group_bracket": "[5,5]", "order": 25, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "-2", "4", "2", "5", "2", "1" ], "h": [ "1", "1", "1", "1" ], "f_str": "x^5 + 2*x^4 + 5*x^3 + 2*x^2 + 4*x - 2", "h_str": "x^3 + x^2 + x + 1", "g": [ "-7", "18", "11", "24", "11", "6", "1" ], "g_str": "x^6 + 6*x^5 + 11*x^4 + 24*x^3 + 11*x^2 + 18*x - 7", "g_scale": "1", "discriminant": "245635219456", "discriminant_digits": 12, "igusa_clebsch": [ "7520", "1183744", "3001843712", "245635219456" ], "g2_invariants": [ "716703146875/58564", "14444374875/42592", "-717925/704" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "-2", "4", "2", "5", "2", "1" ], "h": [ "1", "1", "1", "1" ] }, "minimal_discriminant": "59969536", "minimal_discriminant_factored": "2^12 * 11^4", "conductor": "121", "conductor_factored": "11 * 11 (product of the elliptic conductors)", "conductor_note": "Magma's genus-2 Conductor gave 123904 (Ogg's formula at 2, no correctness guarantee); the rigorous value is N(E1) N(E2) = 11 * 11 for the elliptic curves ['0', '-1', '1', '0', '0'] and ['0', '-1', '1', '-10', '-20'] to whose product J is isogenous over Q", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "123904" }, "lmfdb": { "label": "121.a.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-2%2C4%2C2%2C5%2C2%2C1%5D%2C%5B1%2C1%2C1%2C1%5D%5D", "snapshot_label": "121.a.1" }, "torsion": { "invariants": [ 5, 5 ], "order": 25, "generators": [ { "order": 5, "mumford": [ "x^2 + x + 2", "-x - 3", "2" ] }, { "order": 5, "mumford": [ "x^2 + 1/3*x + 4/3", "11/9*x + 5/9", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 25, "point_count_primes": [ 3, 5, 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 5, 5 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 93, "reducible_chi": 93 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x + 3*z : 2*x^3 - 2*x^2*z - 10*x*z^2 + 8*y - 6*z^3 : x - z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "0", "-1", "1", "0", "0" ], [ "0", "-1", "1", "-10", "-20" ] ], "conductors": [ "11", "11" ], "conductor": "121", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 44, "reducible_chi": 44 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Howe–Leprévost–Poonen 2000", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 55 (source: \\cite{HLP2000})", "sources": [ "HLP2000" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-2", "4", "2", "5", "2", "1" ], "h": [ "1", "1", "1", "1" ], "claimed_group": [ 5, 5 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "121.a.1" }, "source": { "kind": "import", "name": "paper-t2-055", "file": "table2.txt", "line": 61, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:38:07 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.19000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/5.5.a.log", "conductor_log": "data/logs/5.5.a.conductor.log", "conductor_cputime_seconds": 0.19 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "5.5.b", "group": [ 5, 5 ], "group_key": "5.5", "group_bracket": "[5,5]", "order": 25, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "428", "-264", "295", "75", "121", "45", "4" ], "h": [ "0", "0", "0", "1" ], "f_str": "4*x^6 + 45*x^5 + 121*x^4 + 75*x^3 + 295*x^2 - 264*x + 428", "h_str": "x^3", "g": [ "1712", "-1056", "1180", "300", "484", "180", "17" ], "g_str": "17*x^6 + 180*x^5 + 484*x^4 + 300*x^3 + 1180*x^2 - 1056*x + 1712", "g_scale": "1", "discriminant": "-3726636128673542353918623744", "discriminant_digits": 28, "igusa_clebsch": [ "-23186080", "20319452591104", "-134251546924553146368", "-3726636128673542353918623744" ], "g2_invariants": [ "9736363260162577365625/43317870228864", "34800018676231693841125/9401375190316032", "-2236111235794087285825/38330323338295296" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "428", "-264", "295", "75", "121", "45", "4" ], "h": [ "0", "0", "0", "1" ] }, "minimal_discriminant": "-909823273601939051249664", "minimal_discriminant_factored": "-1 * 2^17 * 3^10 * 7^5 * 11 * 29^5 * 31", "conductor": "1275918336", "conductor_factored": "2^11 * 3^2 * 7 * 11 * 29 * 31", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B428%2C-264%2C295%2C75%2C121%2C45%2C4%5D%2C%5B0%2C0%2C0%2C1%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 5, 5 ], "order": 25, "generators": [ { "order": 5, "mumford": [ "x^2 + 3/4*x - 53/2", "-1569/16*x - 4113/8", "2" ] }, { "order": 5, "mumford": [ "x^2 + 36*x - 28", "4674*x - 3516", "2" ] } ], "method": "generators", "generators_note": "", "generators_submitted": 2, "point_count_gcd": 25, "point_count_primes": [ 5, 13, 19, 23, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131 ], "claimed": [ 5, 5 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 23, "chi": "T^4 + 2*T^3 - 3*T^2 + 46*T + 529", "primes_tried_below": 500, "admissible_primes": 4, "reducible_chi": 3 }, "q_simple": { "certified": true, "prime": 23, "chi": "T^4 + 2*T^3 - 3*T^2 + 46*T + 529" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 39, "reducible_chi": 14 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "F. Najman", "year": 2026, "historical": false }, "reference": "new", "sources": [], "rm": false, "notes": "First example of [5,5] = Z/5 x Z/5 for a geometrically simple genus 2 Jacobian over Q (the smallest group listed as unrealized in arXiv:2608.28543, Open Question 1).\n\nOutput of pipeline/magma/prepare_submission.m (Magma V2.29-4): J(Q)_tors = [5,5]; gcd of #J(F_p) over 25 good primes = 25 (sharp); strict prime 23, chi = x^4 + 2*x^3 - 3*x^2 + 46*x + 529. Also strict at p = 17 (chi = x^4 - x^3 + 3*x^2 - 17*x + 289), with 25 strict primes below 200. Independent PARI/GP check (hyperellcharpoly): 25 | #J(F_p) for all 71 good odd p < 400, gcd 25. No RM over Q (the squarefree part of a1^2 - 4(a2 - 2p) varies with p), so End_Q(J) = Z. #Aut(C) = 2.\n\nConductor: the odd part is 623007 = 3^2*7*11*29*31 (PARI genus2red; Neron component groups [5,5] at 3, [5] at 7 and 29). At 2, Magma's Conductor gives 2^11 (N = 1275918336) using Ogg's formula, which Magma does not guarantee here because v_2(disc) = 17.\n\nHow it was found: by a sieve over Elkies' universal 5-torsion family y^2 + H y = Q^2 Q', with Q = q2 x^2 + q1 x + q0, Q' = Q - x and H = Q' - x Q. The sieve kept parameters with Frobenius polynomial = (x-1)^2 (x-p)^2 mod 5 and 25 | #J(F_p) at good primes. The curve is at (q0,q1,q2) = (21/50, -23/50, 3/25). All other [5,5] points the sieve found at that height are bielliptic and split over Q.", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "428", "-264", "295", "75", "121", "45", "4" ], "h": [ "0", "0", "0", "1" ], "claimed_group": [ 5, 5 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "issue", "number": 1, "url": "https://github.com/Genus-2-torsion/Genus-2-torsion.github.io/issues/1" }, "dates": { "submitted": "2026-10-12", "verified": "2026-10-12 15:04:49 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.200000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/5.5.b.log", "conductor_log": "data/logs/5.5.b.conductor.log", "conductor_cputime_seconds": 0.44 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "5.10.a", "group": [ 5, 10 ], "group_key": "5.10", "group_bracket": "[5,10]", "order": 50, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "-135", "135", "11", "-33", "15", "-3", "9" ], "h": [ "0", "1", "1" ], "f_str": "9*x^6 - 3*x^5 + 15*x^4 - 33*x^3 + 11*x^2 + 135*x - 135", "h_str": "x^2 + x", "g": [ "-540", "540", "45", "-130", "61", "-12", "36" ], "g_str": "36*x^6 - 12*x^5 + 61*x^4 - 130*x^3 + 45*x^2 + 540*x - 540", "g_scale": "1", "discriminant": "256083186995514939801600000", "discriminant_digits": 27, "igusa_clebsch": [ "4463880", "675449067780", "867367546876341000", "256083186995514939801600000" ], "g2_invariants": [ "71228914465366641785999/82331271539195904", "636664218571529723911/38592783533998080", "-7488309819263545/114254951251968" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "-135", "135", "11", "-33", "15", "-3", "9" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "62520309325076889600000", "minimal_discriminant_factored": "2^25 * 3^22 * 5^5 * 19", "conductor": "85500", "conductor_factored": "150 * 570 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 150 * 570 for the elliptic curves ['1', '0', '0', '-28', '272'] and ['1', '0', '0', '9335', '-737383'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "85500" }, "lmfdb": { "label": "85500.cj.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-135%2C135%2C11%2C-33%2C15%2C-3%2C9%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "85500.cj.1" }, "torsion": { "invariants": [ 5, 10 ], "order": 50, "generators": [ { "order": 5, "mumford": [ "x^2 + 5/6*x - 5/2", "-23*x + 15", "2" ] }, { "order": 10, "mumford": [ "x - 3", "-6*x^3 + 324", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 100, "point_count_primes": [ 7, 11, 13, 17, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 5, 10 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 91, "reducible_chi": 91 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x : -2/9*x^3 + 4/3*x^2*z + y : 1/3*x - z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "0", "0", "-28", "272" ], [ "1", "0", "0", "9335", "-737383" ] ], "conductors": [ "150", "570" ], "conductor": "85500", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 42, "reducible_chi": 42 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 56 (source: \\cite{HLP2000})", "sources": [ "HLP2000" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-135", "135", "11", "-33", "15", "-3", "9" ], "h": [ "0", "1", "1" ], "claimed_group": [ 5, 10 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "85500.cj.1" }, "source": { "kind": "import", "name": "paper-t2-056", "file": "table2.txt", "line": 62, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:38:09 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.20000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/5.10.a.log", "conductor_log": "data/logs/5.10.a.conductor.log", "conductor_cputime_seconds": 0.48 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "6.6.a", "group": [ 6, 6 ], "group_key": "6.6", "group_bracket": "[6,6]", "order": 36, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "4875", "-1691", "6969", "-3000", "1872" ], "h": [ "0" ], "f_str": "1872*x^5 - 3000*x^4 + 6969*x^3 - 1691*x^2 + 4875*x", "h_str": "0", "g": [ "0", "19500", "-6764", "27876", "-12000", "7488" ], "g_str": "7488*x^5 - 12000*x^4 + 27876*x^3 - 6764*x^2 + 19500*x", "g_scale": "1", "discriminant": "190548703667030661328896000000000000", "discriminant_digits": 36, "igusa_clebsch": [ "9204380256", "1398931521439924224", "5191057458710881596582395904", "10684093253623737233126430081024000000000000" ], "g2_invariants": [ "8102454401252697064182013816427167970801/10482659804599017501492000000000000", "11758999909710518975683219811926427651/496220582466225680544000000000000", "-3381492611643033688726377853/8033149247418880000000000" ], "rational_weierstrass_points": 2, "minimal_model": { "f": [ "0", "4875", "-1691", "6969", "-3000", "1872" ], "h": [] }, "minimal_discriminant": "2608421204497982722931257344000000000000", "minimal_discriminant_factored": "2^24 * 3^9 * 5^12 * 13^6 * 17^3 * 19^6 * 29", "conductor": "17506967961600", "conductor_factored": "2^14 * 3^3 * 5^2 * 13^2 * 17 * 19 * 29", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C4875%2C-1691%2C6969%2C-3000%2C1872%5D%2C%5B0%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 6, 6 ], "order": 36, "generators": [ { "order": 6, "mumford": [ "x^2 - 17/52*x + 15/13", "635/13*x + 30/13", "2" ] }, { "order": 6, "mumford": [ "x^2 + 149/156*x + 25/52", "-1175/13*x - 1200/13", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 36, "point_count_primes": [ 7, 11, 23, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131 ], "claimed": [ 6, 6 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 37, "chi": "T^4 - 10*T^3 + 54*T^2 - 370*T + 1369", "primes_tried_below": 500, "admissible_primes": 5, "reducible_chi": 3 }, "q_simple": { "certified": true, "prime": 23, "chi": "T^4 - 26*T^2 + 529" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 39, "reducible_chi": 19 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Balakrishnan–Najman–Shnidman–Sutherland 2026", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 56 (source: new)", "sources": [ "BNSS2026" ], "rm": false, "notes": "new (this paper)", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "4875", "-1691", "6969", "-3000", "1872" ], "h": [ "0" ], "claimed_group": [ 6, 6 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "import", "name": "paper-t1-056", "file": "table1.txt", "line": 62, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:54 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.360000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/6.6.a.log", "conductor_log": "data/logs/6.6.a.conductor.log", "conductor_cputime_seconds": 0.51 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "6.6.b", "group": [ 6, 6 ], "group_key": "6.6", "group_bracket": "[6,6]", "order": 36, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "1", "3", "6", "7", "6", "3", "1" ], "h": [ "0", "1", "1" ], "f_str": "x^6 + 3*x^5 + 6*x^4 + 7*x^3 + 6*x^2 + 3*x + 1", "h_str": "x^2 + x", "g": [ "4", "12", "25", "30", "25", "12", "4" ], "g_str": "4*x^6 + 12*x^5 + 25*x^4 + 30*x^3 + 25*x^2 + 12*x + 4", "g_scale": "1", "discriminant": "-89915392", "discriminant_digits": 8, "igusa_clebsch": [ "-2680", "5380", "-1198840", "-89915392" ], "g2_invariants": [ "4219140959375/21952", "6203236875/784", "12905875/28" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "1", "3", "6", "7", "6", "3", "1" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "-21952", "minimal_discriminant_factored": "-1 * 2^6 * 7^3", "conductor": "196", "conductor_factored": "14 * 14 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 14 * 14 for the elliptic curves ['1', '0', '1', '-1', '0'] and ['1', '0', '1', '4', '-6'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "196" }, "lmfdb": { "label": "196.a.21952.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/196/a/21952/1", "snapshot_label": "" }, "torsion": { "invariants": [ 6, 6 ], "order": 36, "generators": [ { "order": 6, "mumford": [ "x^2 + 1/2*x + 3/2", "2", "2" ] }, { "order": 6, "mumford": [ "x^2 + 3/4*x + 1/2", "1/8*x + 3/4", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 36, "point_count_primes": [ 3, 5, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 6, 6 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 93, "reducible_chi": 93 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x : y : -x - z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "0", "1", "-1", "0" ], [ "1", "0", "1", "4", "-6" ] ], "conductors": [ "14", "14" ], "conductor": "196", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 44, "reducible_chi": 44 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 57 (source: \\cite{HLP2000})", "sources": [ "HLP2000" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "1", "3", "6", "7", "6", "3", "1" ], "h": [ "0", "1", "1" ], "claimed_group": [ 6, 6 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "196.a.21952.1" }, "source": { "kind": "import", "name": "paper-t2-057", "file": "table2.txt", "line": 63, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:38:11 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.18000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/6.6.b.log", "conductor_log": "data/logs/6.6.b.conductor.log", "conductor_cputime_seconds": 0.13 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "6.12.a", "group": [ 6, 12 ], "group_key": "6.12", "group_bracket": "[6,12]", "order": 72, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "88825", "81950", "75207", "-13354", "-6347", "396", "132" ], "h": [ "0" ], "f_str": "132*x^6 + 396*x^5 - 6347*x^4 - 13354*x^3 + 75207*x^2 + 81950*x + 88825", "h_str": "0", "g": [ "355300", "327800", "300828", "-53416", "-25388", "1584", "528" ], "g_str": "528*x^6 + 1584*x^5 - 25388*x^4 - 53416*x^3 + 300828*x^2 + 327800*x + 355300", "g_scale": "1", "discriminant": "-6797932297474446086442619882687054762475520000", "discriminant_digits": 46, "igusa_clebsch": [ "115064146560", "1798962336358103245824", "43422974416595413301150668455936", "-6797932297474446086442619882687054762475520000" ], "g2_invariants": [ "-149017444812031951377933183215/401794355360745414144", "506216217164521430702089441/27902385788940653760", "-560580463801097781323/706146384762225" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "22206", "20487", "18801", "-3339", "-1587", "99", "33" ], "h": [ "1", "1", "1" ] }, "minimal_discriminant": "-1582766952336404027054696316525120000", "minimal_discriminant_factored": "-1 * 2^9 * 3^11 * 5^4 * 7^3 * 11^22", "conductor": "18900", "conductor_factored": "630 * 30 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 630 * 30 for the elliptic curves ['1', '-1', '1', '-197573', '33848381'] and ['1', '0', '1', '-19', '26'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "18900" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B88825%2C81950%2C75207%2C-13354%2C-6347%2C396%2C132%5D%2C%5B0%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 6, 12 ], "order": 72, "generators": [ { "order": 6, "mumford": [ "x^2 + x + 15/4", "605/2*x + 605/4", "2" ] }, { "order": 12, "mumford": [ "x^2 + 35/2*x + 34", "-2541*x - 6171", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 144, "point_count_primes": [ 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113 ], "claimed": [ 6, 12 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 90, "reducible_chi": 90 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x + z : y : -z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "-1", "1", "-197573", "33848381" ], [ "1", "0", "1", "-19", "26" ] ], "conductors": [ "630", "30" ], "conductor": "18900", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 41, "reducible_chi": 41 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Balakrishnan–Najman–Shnidman–Sutherland 2026", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 58 (source: \\cite{HLP2000})", "sources": [ "HLP2000" ], "rm": false, "notes": "new: glue 30.a6 x cond-630", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "88825", "81950", "75207", "-13354", "-6347", "396", "132" ], "h": [ "0" ], "claimed_group": [ 6, 12 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "import", "name": "paper-t2-058", "file": "table2.txt", "line": 64, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:38:12 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.980000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/6.12.a.log", "conductor_log": "data/logs/6.12.a.conductor.log", "conductor_cputime_seconds": 0.41 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "7.7.a", "group": [ 7, 7 ], "group_key": "7.7", "group_bracket": "[7,7]", "order": 49, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "869675859", "0", "3232987", "0", "3025", "0", "1" ], "h": [ "0" ], "f_str": "x^6 + 3025*x^4 + 3232987*x^2 + 869675859", "h_str": "0", "g": [ "3478703436", "0", "12931948", "0", "12100", "0", "4" ], "g_str": "4*x^6 + 12100*x^4 + 12931948*x^2 + 3478703436", "g_scale": "1", "discriminant": "-576386630760014045574471303713805712755962612469989376", "discriminant_digits": 54, "igusa_clebsch": [ "-5843180431360", "25272297076858273398784", "-35073570663235318199727678451351552", "-576386630760014045574471303713805712755962612469989376" ], "g2_invariants": [ "189058724640202566877858229103396357889920153125/127983541711177216798505385789443014656", "2190328392680325235597490832053495905479625/36012392588703096422498086114295808", "280844873092297136402204005615675/82817965374161539011969024" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "13639248", "202440", "203197", "1515", "760", "3", "1" ], "h": [] }, "minimal_discriminant": "-131055146712245470001669515048389647007744", "minimal_discriminant_factored": "-1 * 2^38 * 3^21 * 7^14 * 13^5 * 181", "conductor": "53958996", "conductor_factored": "98826 * 546 (product of the elliptic conductors)", "conductor_note": "Magma's genus-2 Conductor gave 56580108189696 (Ogg's formula at 2, no correctness guarantee); the rigorous value is N(E1) N(E2) = 98826 * 546 for the elliptic curves ['1', '0', '0', '-3333198596', '103601868154512'] and ['1', '0', '0', '714', '-82908'] to whose product J is isogenous over Q", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "56580108189696" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B869675859%2C0%2C3232987%2C0%2C3025%2C0%2C1%5D%2C%5B0%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 7, 7 ], "order": 49, "generators": [ { "order": 7, "mumford": [ "x^2 - 2353", "-4032*x", "2" ] }, { "order": 7, "mumford": [ "x^2 + 14*x + 1329", "420*x + 1596", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 49, "point_count_primes": [ 5, 11, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 7, 7 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 90, "reducible_chi": 90 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x : y : -z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "0", "0", "-3333198596", "103601868154512" ], [ "1", "0", "0", "714", "-82908" ] ], "conductors": [ "98826", "546" ], "conductor": "53958996", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 41, "reducible_chi": 41 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Howe–Leprévost–Poonen 2000", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 59 (source: \\cite{HLP2000})", "sources": [ "HLP2000" ], "rm": false, "notes": "HLP sec 3.6 curve / lane8", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "869675859", "0", "3232987", "0", "3025", "0", "1" ], "h": [ "0" ], "claimed_group": [ 7, 7 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "import", "name": "paper-t2-059", "file": "table2.txt", "line": 65, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:38:14 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.08000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/7.7.a.log", "conductor_log": "data/logs/7.7.a.conductor.log", "conductor_cputime_seconds": 1.02 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "8.8.a", "group": [ 8, 8 ], "group_key": "8.8", "group_bracket": "[8,8]", "order": 64, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "84285504", "-4535664", "-4045487", "1800118", "88597", "88596", "836" ], "h": [ "0" ], "f_str": "836*x^6 + 88596*x^5 + 88597*x^4 + 1800118*x^3 - 4045487*x^2 - 4535664*x + 84285504", "h_str": "0", "g": [ "337142016", "-18142656", "-16181948", "7200472", "354388", "354384", "3344" ], "g_str": "3344*x^6 + 354384*x^5 + 354388*x^4 + 7200472*x^3 - 16181948*x^2 - 18142656*x + 337142016", "g_scale": "1", "discriminant": "225751817148534949760712850840798633254363529714010402240593920000000000", "discriminant_digits": 72, "igusa_clebsch": [ "-124919583375232", "-256220873146830505550742158336", "83389359739313324664316761318160242961317888", "225751817148534949760712850840798633254363529714010402240593920000000000" ], "g2_invariants": [ "-885325120877126740813568276797172194693334480076303736761599/52561940892724075764583621835522035428873153802475520000000000", "-24319519833008949334758072760730520324935595445301246293657/131404852231810189411459054588805088572182884506188800000000", "-90592285778673328388100193839690849341726595097/153730349421064657152215506200451321989184000000" ], "rational_weierstrass_points": 0, "minimal_model": { "f": [ "21071376", "-1133916", "-1011372", "450029", "22149", "22149", "209" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "52561940892724075764583621835522035428873153802475520000000000", "minimal_discriminant_factored": "2^20 * 3^8 * 5^10 * 7^2 * 17^2 * 23 * 1913^12", "conductor": "17243100", "conductor_factored": "82110 * 210 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 82110 * 210 for the elliptic curves ['1', '0', '0', '-49423080', '130545230400'] and ['1', '0', '0', '-1070', '7812'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "17243100" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B84285504%2C-4535664%2C-4045487%2C1800118%2C88597%2C88596%2C836%5D%2C%5B0%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 8, 8 ], "order": 64, "generators": [ { "order": 8, "mumford": [ "x^2 + 195/47*x + 1638/47", "4391240/2209*x - 8192130/2209", "2" ] }, { "order": 8, "mumford": [ "x^2 - 1751/604*x - 1035/604", "133903839/182408*x + 1752282519/182408", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 256, "point_count_primes": [ 13, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137 ], "claimed": [ 8, 8 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 87, "reducible_chi": 87 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x - 8151/925*z : 7000755497/791453125*y : -344/925*x - z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "0", "0", "-49423080", "130545230400" ], [ "1", "0", "0", "-1070", "7812" ] ], "conductors": [ "82110", "210" ], "conductor": "17243100", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 38, "reducible_chi": 38 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Balakrishnan–Najman–Shnidman–Sutherland 2026", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 60 (source: \\cite{HLP2000})", "sources": [ "HLP2000" ], "rm": false, "notes": "new: glue 82110.bs5 x 210.e6", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "84285504", "-4535664", "-4045487", "1800118", "88597", "88596", "836" ], "h": [ "0" ], "claimed_group": [ 8, 8 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "import", "name": "paper-t2-060", "file": "table2.txt", "line": 66, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:38:16 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.24000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/8.8.a.log", "conductor_log": "data/logs/8.8.a.conductor.log", "conductor_cputime_seconds": 0.3 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.2.a", "group": [ 2, 2, 2 ], "group_key": "2.2.2", "group_bracket": "[2,2,2]", "order": 8, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "1", "-3", "-5", "8", "10" ], "h": [ "0", "1" ], "f_str": "10*x^5 + 8*x^4 - 5*x^3 - 3*x^2 + x", "h_str": "x", "g": [ "0", "4", "-11", "-20", "32", "40" ], "g_str": "40*x^5 + 32*x^4 - 20*x^3 - 11*x^2 + 4*x", "g_scale": "1", "discriminant": "865280", "discriminant_digits": 6, "igusa_clebsch": [ "14432", "2670976", "13424043776", "1384448000" ], "g2_invariants": [ "1194160449744064/21125", "39547563972312/21125", "859738601216/21125" ], "rational_weierstrass_points": 4, "minimal_model": { "f": [ "0", "-10", "8", "5", "-3", "-1" ], "h": [ "0", "0", "1" ] }, "minimal_discriminant": "338000", "minimal_discriminant_factored": "2^4 * 5^3 * 13^2", "conductor": "2600", "conductor_factored": "2^3 * 5^2 * 13", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "2600.a.338000.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/2600/a/338000/1", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 2, 2 ], "order": 8, "generators": [ { "order": 2, "mumford": [ "x", "0", "1" ] }, { "order": 2, "mumford": [ "x + 4/5", "0", "1" ] }, { "order": 2, "mumford": [ "x^2 + 1/2*x - 1/4", "0", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 8, "point_count_primes": [ 3, 7, 11, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 2, 2, 2 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 11, "chi": "T^4 + 4*T^3 + 6*T^2 + 44*T + 121", "primes_tried_below": 500, "admissible_primes": 3, "reducible_chi": 1 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 - 2*T^2 + 9" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 43, "reducible_chi": 20 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 57 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "1", "-3", "-5", "8", "10" ], "h": [ "0", "1" ], "claimed_group": [ 2, 2, 2 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "2600.a.338000.1" }, "source": { "kind": "import", "name": "paper-t1-057", "file": "table1.txt", "line": 63, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:55 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.340000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.2.a.log", "conductor_log": "data/logs/2.2.2.a.conductor.log", "conductor_cputime_seconds": 0.19 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.2.b", "group": [ 2, 2, 2 ], "group_key": "2.2.2", "group_bracket": "[2,2,2]", "order": 8, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "0", "1", "32", "268", "129", "16" ], "h": [ "0", "1" ], "f_str": "16*x^5 + 129*x^4 + 268*x^3 + 32*x^2 + x", "h_str": "x", "g": [ "0", "4", "129", "1072", "516", "64" ], "g_str": "64*x^5 + 516*x^4 + 1072*x^3 + 129*x^2 + 4*x", "g_scale": "1", "discriminant": "243855360", "discriminant_digits": 9, "igusa_clebsch": [ "5840320", "975925312", "1889760358709760", "998831554560" ], "g2_invariants": [ "10125193457354153581255000/11907", "1686759712885462241308475/47628", "1965752701247131775" ], "rational_weierstrass_points": 4, "minimal_model": { "f": [ "0", "-16", "129", "-268", "32", "-1" ], "h": [ "0", "0", "1" ] }, "minimal_discriminant": "243855360", "minimal_discriminant_factored": "2^12 * 3^5 * 5 * 7^2", "conductor": "315", "conductor_factored": "15 * 21 (product of the elliptic conductors)", "conductor_note": "Magma's genus-2 Conductor gave 322560 (Ogg's formula at 2, no correctness guarantee); the rigorous value is N(E1) N(E2) = 15 * 21 for the elliptic curves ['1', '1', '1', '-135', '-660'] and ['1', '0', '0', '-49', '-136'] to whose product J is isogenous over Q", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "322560" }, "lmfdb": { "label": "315.a.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C1%2C32%2C268%2C129%2C16%5D%2C%5B0%2C1%5D%5D", "snapshot_label": "315.a.2" }, "torsion": { "invariants": [ 2, 2, 2 ], "order": 8, "generators": [ { "order": 2, "mumford": [ "x^2 + 4*x", "0", "2" ] }, { "order": 2, "mumford": [ "x", "0", "1" ] }, { "order": 2, "mumford": [ "x^2 + 4*x + 1/4", "0", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 64, "point_count_primes": [ 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 2, 2, 2 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 91, "reducible_chi": 91 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (z : -8*x^2*z + 4*x*z^2 + 8*y : 4*x) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "1", "1", "-135", "-660" ], [ "1", "0", "0", "-49", "-136" ] ], "conductors": [ "15", "21" ], "conductor": "315", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 42, "reducible_chi": 42 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 61 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "1", "32", "268", "129", "16" ], "h": [ "0", "1" ], "claimed_group": [ 2, 2, 2 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "315.a.2" }, "source": { "kind": "import", "name": "paper-t2-061", "file": "table2.txt", "line": 67, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:38:17 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.09000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.2.b.log", "conductor_log": "data/logs/2.2.2.b.conductor.log", "conductor_cputime_seconds": 0.21 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.2.c", "group": [ 2, 2, 2 ], "group_key": "2.2.2", "group_bracket": "[2,2,2]", "order": 8, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "0", "-16", "-8", "23", "-10", "-25" ], "h": [ "0" ], "f_str": "-25*x^5 - 10*x^4 + 23*x^3 - 8*x^2 - 16*x", "h_str": "0", "g": [ "0", "-64", "-32", "92", "-40", "-100" ], "g_str": "-100*x^5 - 40*x^4 + 92*x^3 - 32*x^2 - 64*x", "g_scale": "1", "discriminant": "-13633944184422400", "discriminant_digits": 17, "igusa_clebsch": [ "286304", "-1061273600", "-4546350284800", "-136339441844224000000" ], "g2_invariants": [ "-57330692374283449507/32505856000000", "-23068984144969561569/260046848000000", "-2645395324640711/16777216000000" ], "rational_weierstrass_points": 4, "minimal_model": { "f": [ "0", "25", "-10", "-23", "-8", "16" ], "h": [] }, "minimal_discriminant": "-33285996544000000", "minimal_discriminant_factored": "-1 * 2^36 * 5^6 * 31", "conductor": "775", "conductor_factored": "5^2 * 31", "conductor_note": "taken from the LMFDB label 775.a.2 (Magma did not finish the conductor computation within the time limit)", "conductor_source": "lmfdb label" }, "lmfdb": { "label": "775.a.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C-16%2C-8%2C23%2C-10%2C-25%5D%2C%5B0%5D%5D", "snapshot_label": "775.a.2" }, "torsion": { "invariants": [ 2, 2, 2 ], "order": 8, "generators": [ { "order": 2, "mumford": [ "x + 4/5", "0", "1" ] }, { "order": 2, "mumford": [ "x + 1", "0", "1" ] }, { "order": 2, "mumford": [ "x^2 + 4/5*x", "0", "2" ] } ], "method": "generators", "generators_note": "", "generators_submitted": 3, "point_count_gcd": 8, "point_count_primes": [ 3, 7, 11, 13, 17, 19, 23, 29, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 2, 2, 2 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 50 }, "q_simple": { "certified": true, "prime": 3, "chi": "T^4 - 2*T^2 + 9" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "richelot", "over": "Q", "detail": "(2,2)-isogenous over Q to a Weil restriction of an elliptic curve E over Q(sqrt(5)) (the surface is E x E^sigma)" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 24 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": 2026, "historical": false }, "reference": "LMFDB extended genus 2 database (Booker–Sutherland), curve 775.a.2 (table g2c_curves_new, label 775.a2 on devmirror.lmfdb.xyz, read 2026-10-16)", "sources": [ "BookerSutherland" ], "rm": false, "notes": "First Q-simple, geometrically split example of [2,2,2] for the census (the gsplit cell is empty). Found by querying the LMFDB genus 2 data (devmirror.lmfdb.xyz, g2c_curves_new: is_simple_base = true, is_simple_geom = false, torsion_subgroup = [2,2,2]; geom_end_alg = Q x Q, end_alg = Q; smallest conductor among the curves of this group that the census verifier certifies). LMFDB conductor 775. Checked locally with the census's own pipeline/magma/verify_lib.m (Magma V2.30-1): J(Q)_tors = [2,2,2], class gsplit, Q-simplicity prime 3 (chi = T^4 - 2*T^2 + 9), splitness certificate: richelot over Q ((2,2)-isogenous over Q to a Weil restriction of an elliptic curve E over Q(sqrt(5)) (the surface is E x E^sigma)). Submitted by F. Najman; the curve is due to the database cited under Reference. Rigorous conductor: J is (2,2)-isogenous over Q to Res_(K/Q) E with K the quadratic field of discriminant 5, so N(J) = |d_K|^2 * Norm(f_E) = 25 * 31 = 775 (computed with Magma; conductor is an isogeny invariant).", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "0", "-16", "-8", "23", "-10", "-25" ], "h": [ "0" ], "claimed_group": [ 2, 2, 2 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "775.a.2" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_2.2.2.json" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-17 06:11:58 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "0.770000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.2.c.log", "conductor_log": "data/logs/2.2.2.c.conductor.log", "conductor_cputime_seconds": null }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.4.a", "group": [ 2, 2, 4 ], "group_key": "2.2.4", "group_bracket": "[2,2,4]", "order": 16, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "6", "-8", "-3", "3", "1" ], "h": [ "0", "1", "1" ], "f_str": "x^5 + 3*x^4 - 3*x^3 - 8*x^2 + 6*x", "h_str": "x^2 + x", "g": [ "0", "24", "-31", "-10", "13", "4" ], "g_str": "4*x^5 + 13*x^4 - 10*x^3 - 31*x^2 + 24*x", "g_scale": "1", "discriminant": "238298112", "discriminant_digits": 9, "igusa_clebsch": [ "10888", "1051204", "4056422856", "3812769792" ], "g2_invariants": [ "4669717691462801/930852", "83483209094315/465426", "150912296512/25857" ], "rational_weierstrass_points": 4, "minimal_model": { "f": [ "0", "6", "-8", "-3", "3", "1" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "930852", "minimal_discriminant_factored": "2^2 * 3^4 * 13^2 * 17", "conductor": "3978", "conductor_factored": "2 * 3^2 * 13 * 17", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "3978.a.930852.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/3978/a/930852/1", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 2, 4 ], "order": 16, "generators": [ { "order": 2, "mumford": [ "x", "0", "1" ] }, { "order": 2, "mumford": [ "x + 3", "0", "1" ] }, { "order": 4, "mumford": [ "x^2 + 2*x - 2", "x - 2", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 16, "point_count_primes": [ 5, 7, 11, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 2, 2, 4 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 23, "chi": "T^4 - 2*T^3 + 30*T^2 - 46*T + 529", "primes_tried_below": 500, "admissible_primes": 5, "reducible_chi": 4 }, "q_simple": { "certified": true, "prime": 23, "chi": "T^4 - 2*T^3 + 30*T^2 - 46*T + 529" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 42, "reducible_chi": 13 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 58 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "6", "-8", "-3", "3", "1" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 2, 4 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "3978.a.930852.1" }, "source": { "kind": "import", "name": "paper-t1-058", "file": "table1.txt", "line": 64, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:55 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.340000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.4.a.log", "conductor_log": "data/logs/2.2.4.a.conductor.log", "conductor_cputime_seconds": 0.21 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.4.b", "group": [ 2, 2, 4 ], "group_key": "2.2.4", "group_bracket": "[2,2,4]", "order": 16, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "0", "-15", "7", "18", "0", "3", "-2" ], "h": [ "0", "1", "0", "1" ], "f_str": "-2*x^6 + 3*x^5 + 18*x^3 + 7*x^2 - 15*x", "h_str": "x^3 + x", "g": [ "0", "-60", "29", "72", "2", "12", "-7" ], "g_str": "-7*x^6 + 12*x^5 + 2*x^4 + 72*x^3 + 29*x^2 - 60*x", "g_scale": "1", "discriminant": "-8549802417586176000", "discriminant_digits": 19, "igusa_clebsch": [ "1376", "-365674496", "-2400183648256", "-8549802417586176000" ], "g2_invariants": [ "-147008443/2038431744000", "-50491476899/5435817984000", "-1637394893/28991029248" ], "rational_weierstrass_points": 4, "minimal_model": { "f": [ "0", "-15", "7", "18", "0", "3", "-2" ], "h": [ "0", "1", "0", "1" ] }, "minimal_discriminant": "-2087354105856000", "minimal_discriminant_factored": "-1 * 2^36 * 3^5 * 5^3", "conductor": "225", "conductor_factored": "15 * 15 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 15 * 15 for the elliptic curves ['1', '1', '1', '-5', '2'] and ['1', '1', '1', '-10', '-10'] to whose product J is isogenous over Q", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "" }, "lmfdb": { "label": "225.a.3", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C-15%2C7%2C18%2C0%2C3%2C-2%5D%2C%5B0%2C1%2C0%2C1%5D%5D", "snapshot_label": "225.a.3" }, "torsion": { "invariants": [ 2, 2, 4 ], "order": 16, "generators": [ { "order": 2, "mumford": [ "x^2 - 2*x - 3", "0", "2" ] }, { "order": 2, "mumford": [ "x^2 - 3*x", "0", "2" ] }, { "order": 4, "mumford": [ "x^2 - 5/4*x - 5/4", "103/16*x + 155/16", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 64, "point_count_primes": [ 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 2, 2, 4 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 91, "reducible_chi": 91 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x - 5/7*z : 53/343*x^3 + 243/343*x^2*z - 73/343*x*z^2 + 512/343*y + 185/343*z^3 : -3/7*x - z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "1", "1", "-5", "2" ], [ "1", "1", "1", "-10", "-10" ] ], "conductors": [ "15", "15" ], "conductor": "225", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 42, "reducible_chi": 42 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 62 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "-15", "7", "18", "0", "3", "-2" ], "h": [ "0", "1", "0", "1" ], "claimed_group": [ 2, 2, 4 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "225.a.3" }, "source": { "kind": "import", "name": "paper-t2-062", "file": "table2.txt", "line": 68, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:40:49 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.25000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.4.b.log", "conductor_log": "data/logs/2.2.4.b.conductor.log", "conductor_cputime_seconds": null }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.4.c", "group": [ 2, 2, 4 ], "group_key": "2.2.4", "group_bracket": "[2,2,4]", "order": 16, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "19", "71", "67", "-8", "-19", "4" ], "h": [ "1", "1" ], "f_str": "4*x^5 - 19*x^4 - 8*x^3 + 67*x^2 + 71*x + 19", "h_str": "x + 1", "g": [ "77", "286", "269", "-32", "-76", "16" ], "g_str": "16*x^5 - 76*x^4 - 32*x^3 + 269*x^2 + 286*x + 77", "g_scale": "1", "discriminant": "127704563712", "discriminant_digits": 12, "igusa_clebsch": [ "516288", "299041344", "51809702247936", "32692368310272" ], "g2_invariants": [ "1124718912780739592/8019", "184088347101058945/32076", "265766731402761697/866052" ], "rational_weierstrass_points": 4, "minimal_model": { "f": [ "19", "71", "67", "-8", "-19", "4" ], "h": [ "1", "1" ] }, "minimal_discriminant": "7981535232", "minimal_discriminant_factored": "2^12 * 3^11 * 11", "conductor": "4752", "conductor_factored": "2^4 * 3^3 * 11", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "4752.o.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B19%2C71%2C67%2C-8%2C-19%2C4%5D%2C%5B1%2C1%5D%5D", "snapshot_label": "4752.o.2" }, "torsion": { "invariants": [ 2, 2, 4 ], "order": 16, "generators": [ { "order": 2, "mumford": [ "x^2 - 11/4*x - 11/4", "0", "2" ] }, { "order": 2, "mumford": [ "x - 7/2", "0", "1" ] }, { "order": 4, "mumford": [ "x^2 - 3*x - 1", "3*x - 3", "2" ] } ], "method": "generators", "generators_note": "", "generators_submitted": 3, "point_count_gcd": 16, "point_count_primes": [ 5, 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 2, 2, 4 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 61 }, "q_simple": { "certified": true, "prime": 19, "chi": "T^4 - 10*T^2 + 361" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "richelot", "over": "Q", "detail": "a chain of two (2,2)-isogenies over Q reaches a Weil restriction of an elliptic curve E over Q(sqrt(3)) (the surface is E x E^sigma)" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 29 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": 2026, "historical": false }, "reference": "LMFDB extended genus 2 database (Booker–Sutherland), curve 4752.o.2 (table g2c_curves_new, label 4752.o2 on devmirror.lmfdb.xyz, read 2026-10-16)", "sources": [ "BookerSutherland" ], "rm": false, "notes": "First Q-simple, geometrically split example of [2,2,4] for the census (the gsplit cell is empty). Found by querying the LMFDB genus 2 data (devmirror.lmfdb.xyz, g2c_curves_new: is_simple_base = true, is_simple_geom = false, torsion_subgroup = [2,2,4]; geom_end_alg = Q x Q, end_alg = Q; smallest conductor among the curves of this group that the census verifier certifies). LMFDB conductor 4752. Checked locally with the census's own pipeline/magma/verify_lib.m (Magma V2.30-1): J(Q)_tors = [2,2,4], class gsplit, Q-simplicity prime 19 (chi = T^4 - 10*T^2 + 361), splitness certificate: richelot over Q (a chain of two (2,2)-isogenies over Q reaches a Weil restriction of an elliptic curve E over Q(sqrt(3)) (the surface is E x E^sigma)). Submitted by F. Najman; the curve is due to the database cited under Reference. Rigorous conductor: J is (2,2)-isogenous over Q to Res_(K/Q) E with K the quadratic field of discriminant 12, so N(J) = |d_K|^2 * Norm(f_E) = 144 * 33 = 4752 (computed with Magma; conductor is an isogeny invariant).", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "19", "71", "67", "-8", "-19", "4" ], "h": [ "1", "1" ], "claimed_group": [ 2, 2, 4 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "4752.o.2" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_2.2.4.json" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-17 06:12:03 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "0.940000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.4.c.log", "conductor_log": "data/logs/2.2.4.c.conductor.log", "conductor_cputime_seconds": 3.71 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.6.a", "group": [ 2, 2, 6 ], "group_key": "2.2.6", "group_bracket": "[2,2,6]", "order": 24, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "1", "-1", "-4", "0", "3" ], "h": [ "1", "0", "1" ], "f_str": "3*x^5 - 4*x^3 - x^2 + x", "h_str": "x^2 + 1", "g": [ "1", "4", "-2", "-16", "1", "12" ], "g_str": "12*x^5 + x^4 - 16*x^3 - 2*x^2 + 4*x + 1", "g_scale": "1", "discriminant": "1114112", "discriminant_digits": 7, "igusa_clebsch": [ "3488", "215872", "222549504", "160432128" ], "g2_invariants": [ "61544958196/153", "1836351122/153", "57789184/153" ], "rational_weierstrass_points": 4, "minimal_model": { "f": [ "0", "-3", "0", "4", "-1", "-1" ], "h": [ "0", "1", "0", "1" ] }, "minimal_discriminant": "39168", "minimal_discriminant_factored": "2^8 * 3^2 * 17", "conductor": "816", "conductor_factored": "2^4 * 3 * 17", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "816.a.39168.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/816/a/39168/1", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 2, 6 ], "order": 24, "generators": [ { "order": 2, "mumford": [ "x^2 - 2/3*x - 1/3", "0", "2" ] }, { "order": 2, "mumford": [ "x - 1", "0", "1" ] }, { "order": 6, "mumford": [ "x^2 - 1/3", "2/3", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 24, "point_count_primes": [ 5, 7, 11, 13, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 2, 2, 6 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 29, "chi": "T^4 - 8*T^3 + 46*T^2 - 232*T + 841", "primes_tried_below": 500, "admissible_primes": 7, "reducible_chi": 5 }, "q_simple": { "certified": true, "prime": 5, "chi": "T^4 - 2*T^2 + 25" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 43, "reducible_chi": 18 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 59 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "1", "-1", "-4", "0", "3" ], "h": [ "1", "0", "1" ], "claimed_group": [ 2, 2, 6 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "816.a.39168.1" }, "source": { "kind": "import", "name": "paper-t1-059", "file": "table1.txt", "line": 65, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:56 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.370000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.6.a.log", "conductor_log": "data/logs/2.2.6.a.conductor.log", "conductor_cputime_seconds": 0.19 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.6.b", "group": [ 2, 2, 6 ], "group_key": "2.2.6", "group_bracket": "[2,2,6]", "order": 24, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "0", "1", "-4", "-3", "3", "2" ], "h": [ "0", "1", "0", "1" ], "f_str": "2*x^5 + 3*x^4 - 3*x^3 - 4*x^2 + x", "h_str": "x^3 + x", "g": [ "0", "4", "-15", "-12", "14", "8", "1" ], "g_str": "x^6 + 8*x^5 + 14*x^4 - 12*x^3 - 15*x^2 + 4*x", "g_scale": "1", "discriminant": "73728000", "discriminant_digits": 8, "igusa_clebsch": [ "5504", "381184", "730242816", "73728000" ], "g2_invariants": [ "9634345320448/1125", "320612931584/1125", "289804864/45" ], "rational_weierstrass_points": 4, "minimal_model": { "f": [ "0", "1", "-4", "-3", "3", "2" ], "h": [ "0", "1", "0", "1" ] }, "minimal_discriminant": "18000", "minimal_discriminant_factored": "2^4 * 3^2 * 5^3", "conductor": "600", "conductor_factored": "2^3 * 3 * 5^2", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "600.a.18000.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/600/a/18000/1", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 2, 6 ], "order": 24, "generators": [ { "order": 2, "mumford": [ "x^2 - 1", "0", "2" ] }, { "order": 2, "mumford": [ "x^2 - x", "0", "2" ] }, { "order": 6, "mumford": [ "x", "-x^3", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 72, "point_count_primes": [ 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 2, 2, 6 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "cover", "over": "Q", "detail": "a nonconstant map of degree 3 from y^2 = g to the genus-1 curve v^2 = (1)u^3 + (0)u^2 + (-1728)u + (19008) over Q (the elliptic curve 20a2), (x,y) -> (p/q, y*hh/q^2) with p = 12*x^3 + 60*x^2 + 12*x + 36, q = x^3 + 5*x^2 + 4*x, hh = 216*x^2 + 648*x + 216; degree-3 map to the elliptic curve 20a2 over Q [Genus2Torsion/verify_split_certificates.m, cover 4]" }, "isogenous_product": null, "cover_submitted": true, "cover_accepted": true, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 63 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "1", "-4", "-3", "3", "2" ], "h": [ "0", "1", "0", "1" ], "claimed_group": [ 2, 2, 6 ], "claimed_class": "split", "cover": { "field": "", "cremona": "20a2", "cubic": [], "p": [ "36", "12", "60", "12" ], "q": [ "0", "4", "5", "1" ], "hh": [ "216", "648", "216" ], "note": "degree-3 map to the elliptic curve 20a2 over Q [Genus2Torsion/verify_split_certificates.m, cover 4]" }, "lmfdb_label_given": "600.a.18000.1" }, "source": { "kind": "import", "name": "paper-t2-063", "file": "table2.txt", "line": 69, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:40:50 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.24000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.6.b.log", "conductor_log": "data/logs/2.2.6.b.conductor.log", "conductor_cputime_seconds": 0.13 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.6.c", "group": [ 2, 2, 6 ], "group_key": "2.2.6", "group_bracket": "[2,2,6]", "order": 24, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "1", "-2", "-7", "10", "14", "-8", "-8" ], "h": [ "0" ], "f_str": "-8*x^6 - 8*x^5 + 14*x^4 + 10*x^3 - 7*x^2 - 2*x + 1", "h_str": "0", "g": [ "4", "-8", "-28", "40", "56", "-32", "-32" ], "g_str": "-32*x^6 - 32*x^5 + 56*x^4 + 40*x^3 - 28*x^2 - 8*x + 4", "g_scale": "1", "discriminant": "84537841287168", "discriminant_digits": 14, "igusa_clebsch": [ "75648", "20385792", "524068061184", "84537841287168" ], "g2_invariants": [ "296709280757/81", "93265905227/648", "111265403/16" ], "rational_weierstrass_points": 4, "minimal_model": { "f": [ "-8", "8", "14", "-10", "-7", "2", "1" ], "h": [] }, "minimal_discriminant": "20639121408", "minimal_discriminant_factored": "2^20 * 3^9", "conductor": "6912", "conductor_factored": "2^8 * 3^3", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "6912.w.4", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B1%2C-2%2C-7%2C10%2C14%2C-8%2C-8%5D%2C%5B0%5D%5D", "snapshot_label": "6912.w.4" }, "torsion": { "invariants": [ 2, 2, 6 ], "order": 24, "generators": [ { "order": 2, "mumford": [ "x^2 - 1", "0", "2" ] }, { "order": 2, "mumford": [ "x^2 + 1/2*x - 1/2", "0", "2" ] }, { "order": 6, "mumford": [ "x^2 - x", "-x + 1", "2" ] } ], "method": "generators", "generators_note": "", "generators_submitted": 3, "point_count_gcd": 24, "point_count_primes": [ 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103 ], "claimed": [ 2, 2, 6 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 93, "reducible_chi": 50 }, "q_simple": { "certified": true, "prime": 5, "chi": "T^4 - 2*T^2 + 25" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "richelot", "over": "Q", "detail": "(2,2)-isogenous over Q to a Weil restriction of an elliptic curve E over Q(sqrt(-2)) (the surface is E x E^sigma)" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 44, "reducible_chi": 23 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": 2026, "historical": false }, "reference": "LMFDB extended genus 2 database (Booker–Sutherland), curve 6912.w.4 (table g2c_curves_new, label 6912.w4 on devmirror.lmfdb.xyz, read 2026-10-16)", "sources": [ "BookerSutherland" ], "rm": false, "notes": "First Q-simple, geometrically split example of [2,2,6] for the census (the gsplit cell is empty). Found by querying the LMFDB genus 2 data (devmirror.lmfdb.xyz, g2c_curves_new: is_simple_base = true, is_simple_geom = false, torsion_subgroup = [2,2,6]; geom_end_alg = M_2(Q), end_alg = Q; smallest conductor among the curves of this group that the census verifier certifies). LMFDB conductor 6912. Checked locally with the census's own pipeline/magma/verify_lib.m (Magma V2.30-1): J(Q)_tors = [2,2,6], class gsplit, Q-simplicity prime 5 (chi = T^4 - 2*T^2 + 25), splitness certificate: richelot over Q ((2,2)-isogenous over Q to a Weil restriction of an elliptic curve E over Q(sqrt(-2)) (the surface is E x E^sigma)). Submitted by F. Najman; the curve is due to the database cited under Reference. Rigorous conductor: J is (2,2)-isogenous over Q to Res_(K/Q) E with K the quadratic field of discriminant -8, so N(J) = |d_K|^2 * Norm(f_E) = 64 * 108 = 6912 (computed with Magma; conductor is an isogeny invariant).", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "1", "-2", "-7", "10", "14", "-8", "-8" ], "h": [ "0" ], "claimed_group": [ 2, 2, 6 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "6912.w.4" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_2.2.6.json" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-17 06:12:05 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "1.36000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.6.c.log", "conductor_log": "data/logs/2.2.6.c.conductor.log", "conductor_cputime_seconds": 0.35 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.8.a", "group": [ 2, 2, 8 ], "group_key": "2.2.8", "group_bracket": "[2,2,8]", "order": 32, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "-1", "-6", "-3", "20", "9" ], "h": [ "0", "1", "1" ], "f_str": "9*x^5 + 20*x^4 - 3*x^3 - 6*x^2 - x", "h_str": "x^2 + x", "g": [ "0", "-4", "-23", "-10", "81", "36" ], "g_str": "36*x^5 + 81*x^4 - 10*x^3 - 23*x^2 - 4*x", "g_scale": "1", "discriminant": "17656123392", "discriminant_digits": 11, "igusa_clebsch": [ "24648", "11914884", "87839141256", "22882335916032" ], "g2_invariants": [ "1142489501553907/22989744", "48998854305505/34484616", "1412168220268/38795193" ], "rational_weierstrass_points": 4, "minimal_model": { "f": [ "9", "-21", "3", "10", "-1", "-1" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "5586507792", "minimal_discriminant_factored": "2^4 * 3^14 * 73", "conductor": "3942", "conductor_factored": "2 * 3^3 * 73", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "3942.b.3", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C-1%2C-6%2C-3%2C20%2C9%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "3942.b.3" }, "torsion": { "invariants": [ 2, 2, 8 ], "order": 32, "generators": [ { "order": 2, "mumford": [ "x^2 + 5/3*x - 4/3", "0", "2" ] }, { "order": 2, "mumford": [ "x", "0", "1" ] }, { "order": 8, "mumford": [ "x^2 - 7/9*x - 4/9", "74/9*x + 32/9", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 32, "point_count_primes": [ 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 2, 2, 8 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 7, "chi": "T^4 + 2*T^3 - 2*T^2 + 14*T + 49", "primes_tried_below": 500, "admissible_primes": 2, "reducible_chi": 1 }, "q_simple": { "certified": true, "prime": 7, "chi": "T^4 + 2*T^3 - 2*T^2 + 14*T + 49" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 43, "reducible_chi": 21 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 60 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "-1", "-6", "-3", "20", "9" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 2, 8 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "3942.b.3" }, "source": { "kind": "import", "name": "paper-t1-060", "file": "table1.txt", "line": 66, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:57 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.340000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.8.a.log", "conductor_log": "data/logs/2.2.8.a.conductor.log", "conductor_cputime_seconds": 0.18 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.8.b", "group": [ 2, 2, 8 ], "group_key": "2.2.8", "group_bracket": "[2,2,8]", "order": 32, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "0", "1", "1", "-5", "-4", "6" ], "h": [ "1", "1" ], "f_str": "6*x^5 - 4*x^4 - 5*x^3 + x^2 + x", "h_str": "x + 1", "g": [ "1", "6", "5", "-20", "-16", "24" ], "g_str": "24*x^5 - 16*x^4 - 20*x^3 + 5*x^2 + 6*x + 1", "g_scale": "1", "discriminant": "46080", "discriminant_digits": 5, "igusa_clebsch": [ "9440", "191872", "579060480", "26542080" ], "g2_invariants": [ "28596971960000/81", "1150492082200/81", "6677950400/9" ], "rational_weierstrass_points": 4, "minimal_model": { "f": [ "-5", "0", "7", "0", "-3" ], "h": [ "0", "1", "0", "1" ] }, "minimal_discriminant": "6480", "minimal_discriminant_factored": "2^4 * 3^4 * 5", "conductor": "360", "conductor_factored": "24 * 15 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 24 * 15 for the elliptic curves ['0', '-1', '0', '-4', '4'] and ['1', '1', '1', '-5', '2'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "360" }, "lmfdb": { "label": "360.a.6480.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/360/a/6480/1", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 2, 8 ], "order": 32, "generators": [ { "order": 2, "mumford": [ "x + 1/3", "0", "1" ] }, { "order": 2, "mumford": [ "x^2 + 5/6*x + 1/6", "0", "2" ] }, { "order": 8, "mumford": [ "x", "1", "1" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 64, "point_count_primes": [ 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 2, 2, 8 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x : 2*x^3 + 4*x^2*z + 3*x*z^2 + y + z^3 : -2*x - z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "0", "-1", "0", "-4", "4" ], [ "1", "1", "1", "-5", "2" ] ], "conductors": [ "24", "15" ], "conductor": "360", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "LMFDB genus 2 database (BSSVY 2016)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 64 (source: \\cite{BSSVY})", "sources": [ "BSSVY" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "1", "1", "-5", "-4", "6" ], "h": [ "1", "1" ], "claimed_group": [ 2, 2, 8 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "360.a.6480.1" }, "source": { "kind": "import", "name": "paper-t2-064", "file": "table2.txt", "line": 70, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:40:51 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.890000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.8.b.log", "conductor_log": "data/logs/2.2.8.b.conductor.log", "conductor_cputime_seconds": 0.13 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.8.c", "group": [ 2, 2, 8 ], "group_key": "2.2.8", "group_bracket": "[2,2,8]", "order": 32, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "0", "-25", "46", "5", "-39", "3", "9" ], "h": [ "0", "1", "1" ], "f_str": "9*x^6 + 3*x^5 - 39*x^4 + 5*x^3 + 46*x^2 - 25*x", "h_str": "x^2 + x", "g": [ "0", "-100", "185", "22", "-155", "12", "36" ], "g_str": "36*x^6 + 12*x^5 - 155*x^4 + 22*x^3 + 185*x^2 - 100*x", "g_scale": "1", "discriminant": "175177728000000", "discriminant_digits": 15, "igusa_clebsch": [ "413704", "654796900", "89140378564200", "175177728000000" ], "g2_invariants": [ "369827042982605246689793/42768000000", "14466193255352460790207/42768000000", "78249066161727186361/4752000000" ], "rational_weierstrass_points": 4, "minimal_model": { "f": [ "0", "-25", "-79", "-60", "5", "2" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "42768000000", "minimal_discriminant_factored": "2^10 * 3^5 * 5^6 * 11", "conductor": "9900", "conductor_factored": "2^2 * 3^2 * 5^2 * 11", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "9900.k.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C-25%2C46%2C5%2C-39%2C3%2C9%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "9900.k.1" }, "torsion": { "invariants": [ 2, 2, 8 ], "order": 32, "generators": [ { "order": 2, "mumford": [ "x^2 - 5/6*x", "0", "2" ] }, { "order": 2, "mumford": [ "x^2 + 5/6*x - 25/18", "0", "2" ] }, { "order": 8, "mumford": [ "x^2 + 1/3*x - 25/12", "1/6*x + 25/12", "2" ] } ], "method": "generators", "generators_note": "", "generators_submitted": 3, "point_count_gcd": 32, "point_count_primes": [ 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 2, 2, 8 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 91, "reducible_chi": 63 }, "q_simple": { "certified": true, "prime": 47, "chi": "T^4 - 34*T^2 + 2209" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "richelot", "over": "Q", "detail": "a chain of two (2,2)-isogenies over Q reaches a Weil restriction of an elliptic curve E over Q(sqrt(5)) (the surface is E x E^sigma)" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 42, "reducible_chi": 32 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": 2026, "historical": false }, "reference": "LMFDB extended genus 2 database (Booker–Sutherland), curve 9900.k.1 (table g2c_curves_new, label 9900.k1 on devmirror.lmfdb.xyz, read 2026-10-16)", "sources": [ "BookerSutherland" ], "rm": false, "notes": "First Q-simple, geometrically split example of [2,2,8] for the census (the gsplit cell is empty). Found by querying the LMFDB genus 2 data (devmirror.lmfdb.xyz, g2c_curves_new: is_simple_base = true, is_simple_geom = false, torsion_subgroup = [2,2,8]; geom_end_alg = Q x Q, end_alg = Q; smallest conductor among the curves of this group that the census verifier certifies). LMFDB conductor 9900. Checked locally with the census's own pipeline/magma/verify_lib.m (Magma V2.30-1): J(Q)_tors = [2,2,8], class gsplit, Q-simplicity prime 47 (chi = T^4 - 34*T^2 + 2209), splitness certificate: richelot over Q (a chain of two (2,2)-isogenies over Q reaches a Weil restriction of an elliptic curve E over Q(sqrt(5)) (the surface is E x E^sigma)). Submitted by F. Najman; the curve is due to the database cited under Reference. Rigorous conductor: J is (2,2)-isogenous over Q to Res_(K/Q) E with K the quadratic field of discriminant 5, so N(J) = |d_K|^2 * Norm(f_E) = 25 * 396 = 9900 (computed with Magma; conductor is an isogeny invariant).", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "0", "-25", "46", "5", "-39", "3", "9" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 2, 8 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "9900.k.1" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_2.2.8.json" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-17 06:12:07 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "1.27000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.8.c.log", "conductor_log": "data/logs/2.2.8.c.conductor.log", "conductor_cputime_seconds": 0.3 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.10.a", "group": [ 2, 2, 10 ], "group_key": "2.2.10", "group_bracket": "[2,2,10]", "order": 40, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "0", "-7", "12", "6", "-12" ], "h": [ "1", "0", "1" ], "f_str": "-12*x^5 + 6*x^4 + 12*x^3 - 7*x^2", "h_str": "x^2 + 1", "g": [ "1", "0", "-26", "48", "25", "-48" ], "g_str": "-48*x^5 + 25*x^4 + 48*x^3 - 26*x^2 + 1", "g_scale": "1", "discriminant": "3487629312", "discriminant_digits": 10, "igusa_clebsch": [ "24224", "13526848", "104315471360", "8035497934848" ], "g2_invariants": [ "248587563395557/1915812", "1452789813457/425736", "573049/73" ], "rational_weierstrass_points": 4, "minimal_model": { "f": [ "0", "-12", "7", "9", "-3", "-2" ], "h": [ "0", "0", "1", "1" ] }, "minimal_discriminant": "1961791488", "minimal_discriminant_factored": "2^12 * 3^8 * 73", "conductor": "10512", "conductor_factored": "2^4 * 3^2 * 73", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "10512.n.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C0%2C-7%2C12%2C6%2C-12%5D%2C%5B1%2C0%2C1%5D%5D", "snapshot_label": "10512.n.1" }, "torsion": { "invariants": [ 2, 2, 10 ], "order": 40, "generators": [ { "order": 2, "mumford": [ "x^2 - 1", "0", "2" ] }, { "order": 2, "mumford": [ "x - 1/3", "0", "1" ] }, { "order": 10, "mumford": [ "x^2 - 1/2*x", "7/2*x - 1", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 40, "point_count_primes": [ 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 2, 2, 10 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 17, "chi": "T^4 + 2*T^3 - 6*T^2 + 34*T + 289", "primes_tried_below": 500, "admissible_primes": 5, "reducible_chi": 3 }, "q_simple": { "certified": true, "prime": 7, "chi": "T^4 - 10*T^2 + 49" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 43, "reducible_chi": 18 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 61 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "0", "-7", "12", "6", "-12" ], "h": [ "1", "0", "1" ], "claimed_group": [ 2, 2, 10 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "10512.n.1" }, "source": { "kind": "import", "name": "paper-t1-061", "file": "table1.txt", "line": 67, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:57 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.340000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.10.a.log", "conductor_log": "data/logs/2.2.10.a.conductor.log", "conductor_cputime_seconds": 0.21 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.10.b", "group": [ 2, 2, 10 ], "group_key": "2.2.10", "group_bracket": "[2,2,10]", "order": 40, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-20", "-12", "66", "-2", "-45", "3", "9" ], "h": [ "0", "1", "1" ], "f_str": "9*x^6 + 3*x^5 - 45*x^4 - 2*x^3 + 66*x^2 - 12*x - 20", "h_str": "x^2 + x", "g": [ "-80", "-48", "265", "-6", "-179", "12", "36" ], "g_str": "36*x^6 + 12*x^5 - 179*x^4 - 6*x^3 + 265*x^2 - 48*x - 80", "g_scale": "1", "discriminant": "1346062859108352", "discriminant_digits": 16, "igusa_clebsch": [ "1427336", "1189183108", "560854218306440", "1346062859108352" ], "g2_invariants": [ "180792813437650307954951857/328628627712", "28267728462513508299071/1244805408", "176670550864902399064/142633953" ], "rational_weierstrass_points": 4, "minimal_model": { "f": [ "9", "-57", "105", "-30", "-42", "-6" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "328628627712", "minimal_discriminant_factored": "2^8 * 3^9 * 7^2 * 11^3", "conductor": "30492", "conductor_factored": "2^2 * 3^2 * 7 * 11^2", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "30492.q.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-20%2C-12%2C66%2C-2%2C-45%2C3%2C9%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "30492.q.1" }, "torsion": { "invariants": [ 2, 2, 10 ], "order": 40, "generators": [ { "order": 2, "mumford": [ "x^2 + 1/2*x - 2", "0", "2" ] }, { "order": 2, "mumford": [ "x^2 - 7/3*x + 4/3", "0", "2" ] }, { "order": 10, "mumford": [ "x^2 - 43/18", "-2/3*x - 83/18", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 40, "point_count_primes": [ 5, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 2, 2, 10 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 91, "reducible_chi": 91 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "cover", "over": "Q", "detail": "a nonconstant map of degree 3 from y^2 = g to the genus-1 curve v^2 = (1)u^3 + (0)u^2 + (-58347)u + (3954150) over Q (the elliptic curve 66c1), (x,y) -> (p/q, y*hh/q^2) with p = 3*x^3 - 209/2*x^2 - 4*x + 643/2, q = x^3 + 7/6*x^2 - 4/3*x - 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7/2*x - 2", "0", "2" ] }, { "order": 2, "mumford": [ "x^2 + x", "0", "2" ] }, { "order": 12, "mumford": [ "x^2 - 1", "-3*x - 3", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 48, "point_count_primes": [ 7, 11, 13, 17, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 2, 2, 12 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 19, "chi": "T^4 + 2*T^3 - 18*T^2 + 38*T + 361", "primes_tried_below": 500, "admissible_primes": 5, "reducible_chi": 4 }, "q_simple": { "certified": true, "prime": 19, "chi": "T^4 + 2*T^3 - 18*T^2 + 38*T + 361" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 42, "reducible_chi": 20 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 62 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "-5", "-5", "16", "8", "-7", "1" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 2, 12 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "12300.e.2" }, "source": { "kind": "import", "name": "paper-t1-062", "file": "table1.txt", "line": 68, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:58 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.360000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.12.a.log", "conductor_log": "data/logs/2.2.12.a.conductor.log", "conductor_cputime_seconds": 0.14 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.12.b", "group": [ 2, 2, 12 ], "group_key": "2.2.12", "group_bracket": "[2,2,12]", "order": 48, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "1", "-9", "19", "19", "-55", "-45" ], "h": [ "0", "1", "1" ], "f_str": "-45*x^5 - 55*x^4 + 19*x^3 + 19*x^2 - 9*x + 1", "h_str": "x^2 + x", "g": [ "4", "-36", "77", "78", "-219", "-180" ], "g_str": "-180*x^5 - 219*x^4 + 78*x^3 + 77*x^2 - 36*x + 4", "g_scale": "1", "discriminant": "34560000", "discriminant_digits": 8, "igusa_clebsch": [ "565512", "138678084", "25602606910152", "1119744000000" ], "g2_invariants": [ "7263616663032772854043/1125000", "6011017121094595543/22500", "366293081928308/25" ], "rational_weierstrass_points": 4, "minimal_model": { "f": [ "-69", "58", "44", "-27", "-11", "3", "1" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "273375000", "minimal_discriminant_factored": "2^3 * 3^7 * 5^6", "conductor": "1350", "conductor_factored": "15 * 90 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 15 * 90 for the elliptic curves ['1', '1', '1', '-5', '2'] and ['1', '-1', '1', '-3002', '63929'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "1350" }, "lmfdb": { "label": "1350.d.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B1%2C-9%2C19%2C19%2C-55%2C-45%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "1350.d.1" }, "torsion": { "invariants": [ 2, 2, 12 ], "order": 48, "generators": [ { "order": 2, "mumford": [ "x^2 + 4/5*x - 4/15", "0", "2" ] }, { "order": 2, "mumford": [ "x^2 + 2/3*x - 1/3", "0", "2" ] }, { "order": 12, "mumford": [ "x^2 + 3/4*x - 1/3", "1/2*x", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 96, "point_count_primes": [ 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 2, 2, 12 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x : -6*x^3 + 4*x^2*z + y : 3*x - z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "1", "1", "-5", "2" ], [ "1", "-1", "1", "-3002", "63929" ] ], "conductors": [ "15", "90" ], "conductor": "1350", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 66 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "1", "-9", "19", "19", "-55", "-45" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 2, 12 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "1350.d.1" }, "source": { "kind": "import", "name": "paper-t2-066", "file": "table2.txt", "line": 72, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:40:54 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.08000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.12.b.log", "conductor_log": "data/logs/2.2.12.b.conductor.log", "conductor_cputime_seconds": 0.21 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.14.a", "group": [ 2, 2, 14 ], "group_key": "2.2.14", "group_bracket": "[2,2,14]", "order": 56, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "-150965100", "-169058940", "-16976459", "1218410", "96973", "-4020", "36" ], "h": [ "0" ], "f_str": "36*x^6 - 4020*x^5 + 96973*x^4 + 1218410*x^3 - 16976459*x^2 - 169058940*x - 150965100", "h_str": "0", "g": [ "-603860400", "-676235760", "-67905836", "4873640", "387892", "-16080", "144" ], "g_str": "144*x^6 - 16080*x^5 + 387892*x^4 + 4873640*x^3 - 67905836*x^2 - 676235760*x - 603860400", "g_scale": "1", "discriminant": "256399888026700341452116261526151795195248640000000000000000", "discriminant_digits": 60, "igusa_clebsch": [ "1019780545956992", "8337232088168530324063765504", "2580471846933593342064255001786904824020992", "256399888026700341452116261526151795195248640000000000000000" ], "g2_invariants": [ "867525085403855464231704440432317580274164541967435405644039877/1613452728703695710925361830779878593750000000000", "50016321992370375081993236699411613352462938940602580441/2561036077307453509405336239333140625000000", "84666603759626477552054680936748691862054219573/96216251612940866324986803393750000" ], "rational_weierstrass_points": 4, "minimal_model": { "f": [ "-37741275", "-42264735", "-4244115", "304602", "24243", "-1005", "9" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "59697750962036741304238387738855507968750000000000", "minimal_discriminant_factored": "2^10 * 3^28 * 5^16 * 7^7 * 13^4 * 19^2 * 23^2 * 37 * 317^2", "conductor": "38200383011700", "conductor_factored": "2^2 * 3^2 * 5^2 * 7^2 * 13^2 * 19 * 23 * 37 * 317", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-150965100%2C-169058940%2C-16976459%2C1218410%2C96973%2C-4020%2C36%5D%2C%5B0%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 2, 14 ], "order": 56, "generators": [ { "order": 2, "mumford": [ "x^2 - 145/3*x - 6325/12", "0", "2" ] }, { "order": 2, "mumford": [ "x^2 - 220/3*x + 884", "0", "2" ] }, { "order": 14, "mumford": [ "x^2 - 190/3*x - 718/3", "3285*x + 25860", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 56, "point_count_primes": [ 11, 17, 29, 31, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137 ], "claimed": [ 2, 2, 14 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 17, "chi": "T^4 - 4*T^3 + 6*T^2 - 68*T + 289", "primes_tried_below": 500, "admissible_primes": 2, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 11, "chi": "T^4 - 10*T^2 + 121" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 38, "reducible_chi": 13 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database); Balakrishnan–Najman–Shnidman–Sutherland 2026", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 63 (source: \\cite{BookerSutherland}, new)", "sources": [ "BookerSutherland", "BNSS2026" ], "rm": false, "notes": "new (this paper)", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-150965100", "-169058940", "-16976459", "1218410", "96973", "-4020", "36" ], "h": [ "0" ], "claimed_group": [ 2, 2, 14 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "import", "name": "paper-t1-063", "file": "table1.txt", "line": 69, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:32:59 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.330000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.14.a.log", "conductor_log": "data/logs/2.2.14.a.conductor.log", "conductor_cputime_seconds": 0.37 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.16.a", "group": [ 2, 2, 16 ], "group_key": "2.2.16", "group_bracket": "[2,2,16]", "order": 64, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "9", "-33", "3", "61", "3", "-33", "-11" ], "h": [ "0", "1", "1" ], "f_str": "-11*x^6 - 33*x^5 + 3*x^4 + 61*x^3 + 3*x^2 - 33*x + 9", "h_str": "x^2 + x", "g": [ "36", "-132", "13", "246", "13", "-132", "-44" ], "g_str": "-44*x^6 - 132*x^5 + 13*x^4 + 246*x^3 + 13*x^2 - 132*x + 36", "g_scale": "1", "discriminant": "59595463065600000", "discriminant_digits": 17, "igusa_clebsch": [ "1437512", "708699844", "335368780110152", "59595463065600000" ], "g2_invariants": [ "187330054231141703266877449/14549673600000", "43125489923036794135517/80831520000", "171048761313478133/5832000" ], "rational_weierstrass_points": 4, "minimal_model": { "f": [ "9", "-33", "3", "61", "3", "-33", "-11" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "14549673600000", "minimal_discriminant_factored": "2^10 * 3^10 * 5^5 * 7 * 11", "conductor": "69300", "conductor_factored": "210 * 330 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 210 * 330 for the elliptic curves ['1', '0', '0', '-1070', '7812'] and ['1', '0', '0', '-75', '225'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "69300" }, "lmfdb": { "label": "69300.bu.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B9%2C-33%2C3%2C61%2C3%2C-33%2C-11%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "69300.bu.1" }, "torsion": { "invariants": [ 2, 2, 16 ], "order": 64, "generators": [ { "order": 2, "mumford": [ "x^2 + x - 2", "0", "2" ] }, { "order": 2, "mumford": [ "x^2 + 3/2*x - 1", "0", "2" ] }, { "order": 16, "mumford": [ "x^2 + 2/5*x - 6/5", "-21/25*x - 72/25", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 128, "point_count_primes": [ 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113 ], "claimed": [ 2, 2, 16 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 90, "reducible_chi": 90 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x + z : x^2*z + x*z^2 + y : -z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "0", "0", "-1070", "7812" ], [ "1", "0", "0", "-75", "225" ] ], "conductors": [ "210", "330" ], "conductor": "69300", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 41, "reducible_chi": 41 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 67 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "9", "-33", "3", "61", "3", "-33", "-11" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 2, 16 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "69300.bu.1" }, "source": { "kind": "import", "name": "paper-t2-067", "file": "table2.txt", "line": 73, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:40:56 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.13000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.16.a.log", "conductor_log": "data/logs/2.2.16.a.conductor.log", "conductor_cputime_seconds": 0.33 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.16.b", "group": [ 2, 2, 16 ], "group_key": "2.2.16", "group_bracket": "[2,2,16]", "order": 64, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-284225007", "519681459", "166739244", "-496170642", "128899980", "-11736600", "360000" ], "h": [ "0", "1", "1" ], "f_str": "360000*x^6 - 11736600*x^5 + 128899980*x^4 - 496170642*x^3 + 166739244*x^2 + 519681459*x - 284225007", "h_str": "x^2 + x", "g": [ "-1136900028", "2078725836", "666956977", "-1984682566", "515599921", "-46946400", "1440000" ], "g_str": "1440000*x^6 - 46946400*x^5 + 515599921*x^4 - 1984682566*x^3 + 666956977*x^2 + 2078725836*x - 1136900028", "g_scale": "1", "discriminant": "7772796516032018937652769517842389304044107699223282175230963875840000000000000000", "discriminant_digits": 82, "igusa_clebsch": [ "14621026758469669064", "60760031464812847471078607317270564", "298555470608393610043371064697495313251269388559834536", "7772796516032018937652769517842389304044107699223282175230963875840000000000000000" ], "g2_invariants": [ "20391042259042508183302375667620927548300084866317150945742968724108623464949845362242438393/1897655399421879623450383183066989576182643481255684124812247040000000000000000", "9076064583943568167165093231630465590705693448641207178938247175491738627128223843/20364148107613864908472803802032880337440947129907937280000000000000000", "8241372612628040897570319118865864692112498470152908005060832042539671841/336014882931802695958803457142534742891830439837696000000000000" ], "rational_weierstrass_points": 4, "minimal_model": { "f": [ "-4837915775", "-1775208435", "458065650", "149173042", "-19432020", "-3096600", "360000" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "1897655399421879623450383183066989576182643481255684124812247040000000000000000", "minimal_discriminant_factored": "2^34 * 3^13 * 5^16 * 7^10 * 11^10 * 13^10 * 19^2 * 31 * 37 * 41^2 * 53^2 * 59^2 * 257^2", "conductor": "647548862090857026300", "conductor_factored": "2^2 * 3^2 * 5^2 * 7^2 * 11^2 * 13^2 * 19 * 31 * 37 * 41 * 53 * 59 * 257", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous; computed prime by prime (conductor_by_prime.m), the exponent at 2 on the isomorphic model x -> (0 X + 1)/(1 X + 0)", "conductor_source": "magma" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-284225007%2C519681459%2C166739244%2C-496170642%2C128899980%2C-11736600%2C360000%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 2, 16 ], "order": 64, "generators": [ { "order": 2, "mumford": [ "x^2 - 871/75*x + 506/75", "0", "2" ] }, { "order": 2, "mumford": [ "x^2 + 29/75*x - 46/75", "0", "2" ] }, { "order": 16, "mumford": [ "x^2 - 1131/80*x + 2569/180", "802373/60*x - 568139/45", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 64, "point_count_primes": [ 17, 23, 29, 43, 47, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157 ], "claimed": [ 2, 2, 16 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 47, "chi": "T^4 - 6*T^3 - 2*T^2 - 282*T + 2209", "primes_tried_below": 500, "admissible_primes": 5, "reducible_chi": 3 }, "q_simple": { "certified": true, "prime": 29, "chi": "T^4 - 10*T^2 + 841" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 34, "reducible_chi": 18 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "A. V. Sutherland", "year": 2026, "historical": false }, "reference": "A. V. Sutherland, personal communication to F. Najman (November 2026)", "sources": [], "rm": false, "notes": "First geometrically simple Jacobian with torsion [2,2,16] in the census (the group was known only for Jacobians split over Q, 2.2.16.a). Found by A. V. Sutherland and sent to F. Najman as [[-284225007,519681459,166739244,-496170642,128899980,-11736600,360000],[0,1,1]]. End(J_Qbar) = Z: the Frobenius fields at 47 and 71 are quartic CM fields with no common quadratic subfield (data/knowledge/sources/sutherland_2_2_16_check.log).", "submitter": { "name": "Filip Najman (for A. V. Sutherland)", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "-284225007", "519681459", "166739244", "-496170642", "128899980", "-11736600", "360000" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 2, 16 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "file", "path": "submissions/inbox/sutherland_2_2_16.json" }, "dates": { "submitted": "2026-11-10", "verified": "2026-11-10 08:49:11 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "0.370000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.16.b.log", "conductor_log": "data/logs/2.2.16.b.conductor.log", "conductor_cputime_seconds": 2.83 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.20.a", "group": [ 2, 2, 20 ], "group_key": "2.2.20", "group_bracket": "[2,2,20]", "order": 80, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "1016064", "10501344", "15616273", "-60534098", "27387697", "7579404", "-1566684" ], "h": [ "0" ], "f_str": "-1566684*x^6 + 7579404*x^5 + 27387697*x^4 - 60534098*x^3 + 15616273*x^2 + 10501344*x + 1016064", "h_str": "0", "g": [ "4064256", "42005376", "62465092", "-242136392", "109550788", "30317616", "-6266736" ], "g_str": "-6266736*x^6 + 30317616*x^5 + 109550788*x^4 - 242136392*x^3 + 62465092*x^2 + 42005376*x + 4064256", "g_scale": "1", "discriminant": "93329995430339360332846609274912015463392845314412405612365427507200000000", "discriminant_digits": 74, "igusa_clebsch": [ "299343416200550528", "3972936081196791317070989080781824", "313003616716099412925171939598420022037106389843968", "93329995430339360332846609274912015463392845314412405612365427507200000000" ], "g2_invariants": [ "69951769525324575644013313369977588735925105934655678284695585995995137734001/21730082908258624452363375871188942218058008075322118963200000000", "1860811325832288562697705397355190416862200662421181324822895297690457/47740062411728729504835230122895462663938063547695104000000", "51616751110788211306873419762929693644852509108444176720678401/104882874524586652397067567689120585849262511226880000" ], "rational_weierstrass_points": 4, "minimal_model": { "f": [ "254016", "2625336", "3904068", "-15133525", "6846924", "1894851", "-391671" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "21730082908258624452363375871188942218058008075322118963200000000", "minimal_discriminant_factored": "2^22 * 3^18 * 5^8 * 7^9 * 11^8 * 13^4 * 17^8 * 47^4 * 79 * 227^2", "conductor": "16897247339672700", "conductor_factored": "2^2 * 3^2 * 5^2 * 7^2 * 11^2 * 13 * 17^2 * 47 * 79 * 227", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B1016064%2C10501344%2C15616273%2C-60534098%2C27387697%2C7579404%2C-1566684%5D%2C%5B0%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 2, 20 ], "order": 80, "generators": [ { "order": 2, "mumford": [ "x^2 + 45535/12434*x + 3528/6217", "0", "2" ] }, { "order": 2, "mumford": [ "x^2 + 12265/37302*x + 168/6217", "0", "2" ] }, { "order": 20, "mumford": [ "x^2 - 11426/6217*x - 2016/6217", "-6217*x - 1008", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 80, "point_count_primes": [ 19, 23, 29, 31, 37, 41, 43, 53, 59, 61, 67, 71, 73, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139 ], "claimed": [ 2, 2, 20 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 71, "chi": "T^4 + 2*T^3 + 14*T^2 + 142*T + 5041", "primes_tried_below": 500, "admissible_primes": 12, "reducible_chi": 9 }, "q_simple": { "certified": true, "prime": 43, "chi": "T^4 - 10*T^2 + 1849" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 37, "reducible_chi": 21 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Balakrishnan–Najman–Shnidman–Sutherland 2026", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 64 (source: new)", "sources": [ "BNSS2026" ], "rm": false, "notes": "new (this paper)", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "1016064", "10501344", "15616273", "-60534098", "27387697", "7579404", "-1566684" ], "h": [ "0" ], "claimed_group": [ 2, 2, 20 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "import", "name": "paper-t1-064", "file": "table1.txt", "line": 70, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:33:00 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.450000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.20.a.log", "conductor_log": "data/logs/2.2.20.a.conductor.log", "conductor_cputime_seconds": 0.81 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.24.a", "group": [ 2, 2, 24 ], "group_key": "2.2.24", "group_bracket": "[2,2,24]", "order": 96, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "-158457693276225", "-25400003349776", "19670508870560", "-10404361990400", "4237596000000" ], "h": [ "0" ], "f_str": "4237596000000*x^5 - 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13/60*x - 9/20", "491/30*x - 27/10", "2" ] }, { "order": 4, "mumford": [ "x^2 - 4/5*x + 1", "171/5*x - 14", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 64, "point_count_primes": [ 7, 11, 13, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 2, 4, 4 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 91, "reducible_chi": 91 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x - 15/16*z : -225/32*x^3 + 1455/512*x^2*z + 8001/1024*x*z^2 + 4913/512*y + 15/32*z^3 : -15/4*x - z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "1", "1", "-5", "2" ], [ "1", "1", "1", "-5", "2" ] ], "conductors": [ "15", "15" ], "conductor": "225", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 42, "reducible_chi": 42 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 69 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "15", "40", "0", "161", "-240" ], "h": [ "0", "1" ], "claimed_group": [ 2, 4, 4 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "225.a.5" }, "source": { "kind": "import", "name": "paper-t2-069", "file": "table2.txt", "line": 75, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:40:59 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.03000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.4.4.b.log", "conductor_log": "data/logs/2.4.4.b.conductor.log", "conductor_cputime_seconds": 0.18 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.4.8.a", "group": [ 2, 4, 8 ], "group_key": "2.4.8", "group_bracket": "[2,4,8]", "order": 64, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "-959040", "522729", "4402", "15929", "240" ], "h": [ "0" ], "f_str": "240*x^5 + 15929*x^4 + 4402*x^3 + 522729*x^2 - 959040*x", "h_str": "0", "g": [ "0", "-3836160", "2090916", "17608", "63716", "960" ], "g_str": "960*x^5 + 63716*x^4 + 17608*x^3 + 2090916*x^2 - 3836160*x", "g_scale": "1", "discriminant": "-247332238656445634025221095235050053121867776000", "discriminant_digits": 48, "igusa_clebsch": [ "-2277045155712", "70310637913324367545344", "-69740980751778288614129775497674752", "-227941391145780296317643761368622128957113342361600000" ], "g2_invariants": [ "7331656236901728479007029571641683005280785908693/218402225579549324976268605157225267200000", "8970178033873717670300913989709877336060471187/8190083459233099686610072693395947520000", "-108565959147754679627592399205305764121721/7614747017882014171434984942206976000" ], "rational_weierstrass_points": 4, "minimal_model": { "f": [ "0", "-239760", "130682", "1100", "3982", "60" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "-53071740815830485969233271053205739929600000", "minimal_discriminant_factored": "-1 * 2^28 * 3^22 * 5^5 * 7^12 * 11^4 * 13^2 * 37^2 * 43", "conductor": "10033323300", "conductor_factored": "2^2 * 3^2 * 5^2 * 7^2 * 11 * 13 * 37 * 43", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C-959040%2C522729%2C4402%2C15929%2C240%5D%2C%5B0%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 4, 8 ], "order": 64, "generators": [ { "order": 2, "mumford": [ "x + 333/5", "0", "1" ] }, { "order": 4, "mumford": [ "x^2 + 59/3*x + 296/9", "6496/3*x + 2072", "2" ] }, { "order": 8, "mumford": [ "x^2 + 533/15*x + 768/5", "16366/5*x + 56448/5", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 64, "point_count_primes": [ 17, 19, 23, 29, 31, 41, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137 ], "claimed": [ 2, 4, 8 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 47, "chi": "T^4 - 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10562*x^2 - 1530*x + 7524", "h_str": "x^2 + x", "g": [ "30096", "-6120", "-42247", "3190", "14521", "560" ], "g_str": "560*x^5 + 14521*x^4 + 3190*x^3 - 42247*x^2 - 6120*x + 30096", "g_scale": "1", "discriminant": "6410878680947395822446314520576000", "discriminant_digits": 34, "igusa_clebsch": [ "9739467592", "5685813536617258564", "13943719638201492248546958792", "2010451554345103329919164233652633600000" ], "g2_invariants": [ "2674398679428606946732759324319348293715642249/490832899010034992656045955481600000", "57037102616435192812941129847765989563237/6135411237625437408200574443520000", "-42612318656414027308222478004707/8092501896203225451356672000" ], "rational_weierstrass_points": 4, "minimal_model": { "f": [ "7524", "-1530", "-10562", "797", "3630", "140" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "490832899010034992656045955481600000", "minimal_discriminant_factored": "2^20 * 3^6 * 5^5 * 7^4 * 13 * 37^12", "conductor": "573300", "conductor_factored": "2730 * 210 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 2730 * 210 for the elliptic curves ['1', '1', '1', '-349440', '79361697'] and ['1', '0', '0', '-1070', '7812'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "573300" }, "lmfdb": { "label": "573300.ir.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B7524%2C-1530%2C-10562%2C797%2C3630%2C140%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "573300.ir.1" }, "torsion": { "invariants": [ 2, 4, 8 ], "order": 64, "generators": [ { "order": 2, "mumford": [ "x - 1", "0", "1" ] }, { "order": 4, "mumford": [ "x^2 + 367/35*x - 282/35", "-31914/35*x + 26274/35", "2" ] }, { "order": 8, "mumford": [ "x^2 - 101*x - 116", "27056*x + 30760", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 128, "point_count_primes": [ 11, 17, 19, 23, 29, 31, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127 ], "claimed": [ 2, 4, 8 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 89, "reducible_chi": 89 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x - 63/19*z : 24/361*x^3 + 11892/6859*x^2*z + 50058/6859*x*z^2 + 50653/6859*y + 2583/361*z^3 : -16/19*x - z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "1", "1", "-349440", "79361697" ], [ "1", "0", "0", "-1070", "7812" ] ], "conductors": [ "2730", "210" ], "conductor": "573300", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 40, "reducible_chi": 40 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 70 (source: \\cite{HLP2000})", "sources": [ "HLP2000" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "7524", "-1530", "-10562", "797", "3630", "140" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 4, 8 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "573300.ir.1" }, "source": { "kind": "import", "name": "paper-t2-070", "file": "table2.txt", "line": 76, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:41:01 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.19000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.4.8.b.log", "conductor_log": "data/logs/2.4.8.b.conductor.log", "conductor_cputime_seconds": 0.24 }, "credit_differs": true }, { "schema": "genus-2-torsion/curve/1", "id": "2.6.6.a", "group": [ 2, 6, 6 ], "group_key": "2.6.6", "group_bracket": "[2,6,6]", "order": 72, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "18", "-54", "9", "71", "-40", "-5", "25" ], "h": [ "0", "1", "1" ], "f_str": "25*x^6 - 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5/4", "0", "1" ] }, { "order": 2, "mumford": [ "x + 1/2", "0", "1" ] }, { "order": 2, "mumford": [ "x - 3", "0", "1" ] }, { "order": 2, "mumford": [ "x^2 - 2*x - 3", "0", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 16, "point_count_primes": [ 5, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 2, 2, 2, 2 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 92 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "richelot", "over": "Q", "detail": "(2,2)-isogenous over Q to a Weil restriction of an elliptic curve E over Q(sqrt(7)) (the surface is E x E^sigma)" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 43 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 72 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "26", "41", "-44", "-30", "25", "-4" ], "h": [ "1", "1" ], "claimed_group": [ 2, 2, 2, 2 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "784.c.1" }, "source": { "kind": "import", "name": "paper-t2-072", "file": "table2.txt", "line": 78, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:41:11 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.04000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.2.2.b.log", "conductor_log": "data/logs/2.2.2.2.b.conductor.log", "conductor_cputime_seconds": 7.58 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.2.2.c", "group": [ 2, 2, 2, 2 ], "group_key": "2.2.2.2", "group_bracket": "[2,2,2,2]", "order": 16, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-196", "0", "249", "0", "-54", "0", "1" ], "h": [ "0" ], "f_str": "x^6 - 54*x^4 + 249*x^2 - 196", "h_str": "0", "g": [ "-784", "0", "996", "0", "-216", "0", "4" ], "g_str": "4*x^6 - 216*x^4 + 996*x^2 - 784", "g_scale": "1", "discriminant": "23191881435793368023040000", "discriminant_digits": 26, "igusa_clebsch": [ "4194816", "429983023104", "548982891189633024", "23191881435793368023040000" ], "g2_invariants": [ "151917743390062651/21700507500", "5782636640303227/32550761250", "191650044256/119574225" ], "rational_weierstrass_points": 6, "minimal_model": { "f": [ "0", "9", "-4", "-25", "-10", "3", "1" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "5273223322500", "minimal_discriminant_factored": "2^2 * 3^16 * 5^4 * 7^2", "conductor": "28350", "conductor_factored": "630 * 45 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 630 * 45 for the elliptic curves ['1', '-1', '0', '-630', '4900'] and ['1', '-1', '0', '-45', '-104'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "28350" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-196%2C0%2C249%2C0%2C-54%2C0%2C1%5D%2C%5B0%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 2, 2, 2 ], "order": 16, "generators": [ { "order": 2, "mumford": [ "x^2 - 6*x - 7", "0", "2" ] }, { "order": 2, "mumford": [ "x^2 + 6*x - 7", "0", "2" ] }, { "order": 2, "mumford": [ "x^2 + 3*x + 2", "0", "2" ] }, { "order": 2, "mumford": [ "x^2 - 3*x + 2", "0", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 16, "point_count_primes": [ 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 2, 2, 2, 2 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 91, "reducible_chi": 91 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x : y : -z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "-1", "0", "-630", "4900" ], [ "1", "-1", "0", "-45", "-104" ] ], "conductors": [ "630", "45" ], "conductor": "28350", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 42, "reducible_chi": 42 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "audit of 2026-09-18", "year": 2026, "historical": false }, "reference": "notes/audit_2026-09-18/report.md, item 4: y^2 = (x^2-1)(x^2-4)(x^2-49), split over Q by the involution x -> -x, elliptic quotients of conductors 45 and 630", "sources": [], "rm": false, "notes": "first example of [2,2,2,2] for a Jacobian split over Q in the census; the involution x -> -x gives the two elliptic quotients v^2 = (u-1)(u-4)(u-49) and v^2 = u(u-1)(u-4)(u-49)", "submitter": { "name": "curators (from the audit of 2026-09-18)", "github": "", "affiliation": "" }, "submitted": { "f": [ "-196", "0", "249", "0", "-54", "0", "1" ], "h": [ "0" ], "claimed_group": [ 2, 2, 2, 2 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "file", "path": "submissions/inbox/audit_2222_qsplit.json" }, "dates": { "submitted": "2026-09-18", "verified": "2026-09-18 17:41:21 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.20000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.2.2.c.log", "conductor_log": "data/logs/2.2.2.2.c.conductor.log", "conductor_cputime_seconds": 0.19 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.2.2.d", "group": [ 2, 2, 2, 2 ], "group_key": "2.2.2.2", "group_bracket": "[2,2,2,2]", "order": 16, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "0", "-3", "1", "17", "16", "3" ], "h": [ "0", "1", "1" ], "f_str": "3*x^5 + 16*x^4 + 17*x^3 + x^2 - 3*x", "h_str": "x^2 + x", "g": [ "0", "-12", "5", "70", "65", "12" ], "g_str": "12*x^5 + 65*x^4 + 70*x^3 + 5*x^2 - 12*x", "g_scale": "1", "discriminant": "100088711424", "discriminant_digits": 12, "igusa_clebsch": [ "18440", "15510820", "76221927080", "14412774445056" ], "g2_invariants": [ "65066080869690625/3518743761", "27126047614375/130323843", "-11608959625/14480427" ], "rational_weierstrass_points": 6, "minimal_model": { "f": [ "0", "-3", "1", "17", "16", "3" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "3518743761", "minimal_discriminant_factored": "3^6 * 13^6", "conductor": "1521", "conductor_factored": "3^2 * 13^2", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "1521.b.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C-3%2C1%2C17%2C16%2C3%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "1521.b.1" }, "torsion": { "invariants": [ 2, 2, 2, 2 ], "order": 16, "generators": [ { "order": 2, "mumford": [ "x^2 + 7/4*x + 3/4", "0", "2" ] }, { "order": 2, "mumford": [ "x^2 + 19/4*x + 3", "0", "2" ] }, { "order": 2, "mumford": [ "x", "0", "1" ] }, { "order": 2, "mumford": [ "x + 1", "0", "1" ] } ], "method": "generators", "generators_note": "", "generators_submitted": 4, "point_count_gcd": 16, "point_count_primes": [ 5, 7, 11, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107 ], "claimed": [ 2, 2, 2, 2 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 92, "reducible_chi": 50 }, "q_simple": { "certified": true, "prime": 11, "chi": "T^4 - 10*T^2 + 121" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "richelot", "over": "Q", "detail": "(2,2)-isogenous over Q to a Weil restriction of an elliptic curve E over Q(sqrt(13)) (the surface is E x E^sigma)" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 43, "reducible_chi": 28 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": 2026, "historical": false }, "reference": "LMFDB extended genus 2 database (Booker–Sutherland), curve 1521.b.1 (table g2c_curves_new, label 1521.b1 on devmirror.lmfdb.xyz, read 2026-10-16)", "sources": [ "BookerSutherland" ], "rm": false, "notes": "First Q-simple, geometrically split example of [2,2,2,2] for the census (the gsplit cell is empty). Found by querying the LMFDB genus 2 data (devmirror.lmfdb.xyz, g2c_curves_new: is_simple_base = true, is_simple_geom = false, torsion_subgroup = [2,2,2,2]; geom_end_alg = M_2(Q), end_alg = CM; smallest conductor among the curves of this group that the census verifier certifies). LMFDB conductor 1521. Checked locally with the census's own pipeline/magma/verify_lib.m (Magma V2.30-1): J(Q)_tors = [2,2,2,2], class gsplit, Q-simplicity prime 11 (chi = T^4 - 10*T^2 + 121), splitness certificate: richelot over Q ((2,2)-isogenous over Q to a Weil restriction of an elliptic curve E over Q(sqrt(13)) (the surface is E x E^sigma)). Submitted by F. Najman; the curve is due to the database cited under Reference. Rigorous conductor: J is (2,2)-isogenous over Q to Res_(K/Q) E with K the quadratic field of discriminant 13, so N(J) = |d_K|^2 * Norm(f_E) = 169 * 9 = 1521 (computed with Magma; conductor is an isogeny invariant).", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "0", "-3", "1", "17", "16", "3" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 2, 2, 2 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "1521.b.1" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_2.2.2.2.json" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-17 06:01:54 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "0.880000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.2.2.d.log", "conductor_log": "data/logs/2.2.2.2.d.conductor.log", "conductor_cputime_seconds": 0.0700001 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.2.4.a", "group": [ 2, 2, 2, 4 ], "group_key": "2.2.2.4", "group_bracket": "[2,2,2,4]", "order": 32, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "1", "-6", "-16", "146", "-225" ], "h": [ "0", "1", "1" ], "f_str": "-225*x^5 + 146*x^4 - 16*x^3 - 6*x^2 + x", "h_str": "x^2 + x", "g": [ "0", "4", "-23", "-62", "585", "-900" ], "g_str": "-900*x^5 + 585*x^4 - 62*x^3 - 23*x^2 + 4*x", "g_scale": "1", "discriminant": "914457600", "discriminant_digits": 9, "igusa_clebsch": [ "94344", "47160900", "1443593248200", "740710656000000" ], "g2_invariants": [ "938674930338388651/744187500", "53693607446568211/1116281250", "228605921036416/102515625" ], "rational_weierstrass_points": 6, "minimal_model": { "f": [ "0", "18", "-34", "5", "9", "1" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "180837562500", "minimal_discriminant_factored": "2^2 * 3^10 * 5^6 * 7^2", "conductor": "9450", "conductor_factored": "2 * 3^3 * 5^2 * 7", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "9450.b.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C1%2C-6%2C-16%2C146%2C-225%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "9450.b.2" }, "torsion": { "invariants": [ 2, 2, 2, 4 ], "order": 32, "generators": [ { "order": 2, "mumford": [ "x^2 - 31/60*x + 1/15", "0", "2" ] }, { "order": 2, "mumford": [ "x - 4/15", "0", "1" ] }, { "order": 2, "mumford": [ "x - 1/3", "0", "1" ] }, { "order": 4, "mumford": [ "x^2 + 1/15*x - 1/15", "4*x - 1", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 32, "point_count_primes": [ 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 2, 2, 2, 4 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 17, "chi": "T^4 - 4*T^3 + 6*T^2 - 68*T + 289", "primes_tried_below": 500, "admissible_primes": 3, "reducible_chi": 2 }, "q_simple": { "certified": true, "prime": 17, "chi": "T^4 - 4*T^3 + 6*T^2 - 68*T + 289" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 42, "reducible_chi": 19 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 68 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "1", "-6", "-16", "146", "-225" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 2, 2, 4 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "9450.b.2" }, "source": { "kind": "import", "name": "paper-t1-068", "file": "table1.txt", "line": 74, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:33:03 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.310000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.2.4.a.log", "conductor_log": "data/logs/2.2.2.4.a.conductor.log", "conductor_cputime_seconds": 0.17 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.2.4.b", "group": [ 2, 2, 2, 4 ], "group_key": "2.2.2.4", "group_bracket": "[2,2,2,4]", "order": 32, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "0", "-21", "-6", "34", "16", "-3" ], "h": [ "0", "1", "1" ], "f_str": "-3*x^5 + 16*x^4 + 34*x^3 - 6*x^2 - 21*x", "h_str": "x^2 + x", "g": [ "0", "-84", "-23", "138", "65", "-12" ], "g_str": "-12*x^5 + 65*x^4 + 138*x^3 - 23*x^2 - 84*x", "g_scale": "1", "discriminant": "5402250000000000", "discriminant_digits": 16, "igusa_clebsch": [ "178504", "1210922500", "59223235605000", "777924000000000000" ], "g2_invariants": [ "5530823853093376222793/189922851562500", "45162450609342478691/94961425781250", "1656465165339776/586181640625" ], "rational_weierstrass_points": 6, "minimal_model": { "f": [ "0", "-21", "-6", "34", "16", "-3" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "189922851562500", "minimal_discriminant_factored": "2^2 * 3^4 * 5^12 * 7^4", "conductor": "882", "conductor_factored": "42 * 21 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 42 * 21 for the elliptic curves ['1', '1', '1', '-84', '261'] and ['1', '0', '0', '-4', '-1'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "882" }, "lmfdb": { "label": "882.a.5", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C-21%2C-6%2C34%2C16%2C-3%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "882.a.5" }, "torsion": { "invariants": [ 2, 2, 2, 4 ], "order": 32, "generators": [ { "order": 2, "mumford": [ "x + 1", "0", "1" ] }, { "order": 2, "mumford": [ "x^2 - 17/3*x - 28/3", "0", "2" ] }, { "order": 2, "mumford": [ "x^2 - 31/4*x + 21/4", "0", "2" ] }, { "order": 4, "mumford": [ "x^2 - 4/3*x - 7/9", "49/3*x + 7/3", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 64, "point_count_primes": [ 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 2, 2, 2, 4 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 91, "reducible_chi": 91 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x - 3/4*z : 3/32*x^3 + 3/8*x^2*z + 1/2*x*z^2 + 125/64*y + 21/32*z^3 : -3/4*x - z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "1", "1", "-84", "261" ], [ "1", "0", "0", "-4", "-1" ] ], "conductors": [ "42", "21" ], "conductor": "882", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 42, "reducible_chi": 42 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 73 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "-21", "-6", "34", "16", "-3" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 2, 2, 4 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "882.a.5" }, "source": { "kind": "import", "name": "paper-t2-073", "file": "table2.txt", "line": 79, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:41:12 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.08000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.2.4.b.log", "conductor_log": "data/logs/2.2.2.4.b.conductor.log", "conductor_cputime_seconds": 0.12 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.2.4.c", "group": [ 2, 2, 2, 4 ], "group_key": "2.2.2.4", "group_bracket": "[2,2,2,4]", "order": 32, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "0", "75", "-490", "659", "392", "48" ], "h": [ "0" ], "f_str": "48*x^5 + 392*x^4 + 659*x^3 - 490*x^2 + 75*x", "h_str": "0", "g": [ "0", "300", "-1960", "2636", "1568", "192" ], "g_str": "192*x^5 + 1568*x^4 + 2636*x^3 - 1960*x^2 + 300*x", "g_scale": "1", "discriminant": "10232296799993856000000", "discriminant_digits": 23, "igusa_clebsch": [ "93167456", "4552380006400", "143176481884720332800", "377203389234973507584000000" ], "g2_invariants": [ "209204630614995639704963663037643/89932296093696000000", "2561100354238966139826006650203/26646606249984000000", "87990998318525320573929/16777216000000" ], "rational_weierstrass_points": 6, "minimal_model": { "f": [ "0", "75", "-490", "659", "392", "48" ], "h": [] }, "minimal_discriminant": "92090671199944704000000", "minimal_discriminant_factored": "2^36 * 3^6 * 5^6 * 7^6", "conductor": "176400", "conductor_factored": "2^4 * 3^2 * 5^2 * 7^2", "conductor_note": "taken from the LMFDB label 176400.cr.1: Magma's genus-2 Conductor gave 184968806400, with the exponent of 2 from Ogg's formula (no correctness guarantee when v_2(disc) >= 12); the odd parts agree", "conductor_source": "lmfdb label", "conductor_magma": "184968806400" }, "lmfdb": { "label": "176400.cr.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C75%2C-490%2C659%2C392%2C48%5D%2C%5B0%5D%5D", "snapshot_label": "176400.cr.1" }, "torsion": { "invariants": [ 2, 2, 2, 4 ], "order": 32, "generators": [ { "order": 2, "mumford": [ "x^2 + 5*x", "0", "2" ] }, { "order": 2, "mumford": [ "x", "0", "1" ] }, { "order": 2, "mumford": [ "x^2 - 7/12*x + 1/12", "0", "2" ] }, { "order": 4, "mumford": [ "x^2 - 37/4*x + 15/4", "276*x - 120", "2" ] } ], "method": "generators", "generators_note": "", "generators_submitted": 4, "point_count_gcd": 32, "point_count_primes": [ 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 2, 2, 2, 4 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 91, "reducible_chi": 57 }, "q_simple": { "certified": true, "prime": 19, "chi": "T^4 - 10*T^2 + 361" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "richelot", "over": "Q", "detail": "(2,2)-isogenous over Q to a Weil restriction of an elliptic curve E over Q(sqrt(-5)) (the surface is E x E^sigma)" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 42, "reducible_chi": 28 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": 2026, "historical": false }, "reference": "LMFDB extended genus 2 database (Booker–Sutherland), curve 176400.cr.1 (table g2c_curves_new, label 176400.cr1 on devmirror.lmfdb.xyz, read 2026-10-16)", "sources": [ "BookerSutherland" ], "rm": false, "notes": "First Q-simple, geometrically split example of [2,2,2,4] for the census (the gsplit cell is empty). Found by querying the LMFDB genus 2 data (devmirror.lmfdb.xyz, g2c_curves_new: is_simple_base = true, is_simple_geom = false, torsion_subgroup = [2,2,2,4]; geom_end_alg = Q x Q, end_alg = Q; smallest conductor among the curves of this group that the census verifier certifies). LMFDB conductor 176400. Checked locally with the census's own pipeline/magma/verify_lib.m (Magma V2.30-1): J(Q)_tors = [2,2,2,4], class gsplit, Q-simplicity prime 19 (chi = T^4 - 10*T^2 + 361), splitness certificate: richelot over Q ((2,2)-isogenous over Q to a Weil restriction of an elliptic curve E over Q(sqrt(-5)) (the surface is E x E^sigma)). Note: Magma's genus-2 Conductor on the reduced minimal model gives 184968806400 (2-part by Ogg's formula), not the LMFDB value 176400. Submitted by F. Najman; the curve is due to the database cited under Reference. Rigorous conductor: J is (2,2)-isogenous over Q to Res_(K/Q) E with K the quadratic field of discriminant -20, so N(J) = |d_K|^2 * Norm(f_E) = 400 * 441 = 176400 (computed with Magma; conductor is an isogeny invariant).", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "0", "75", "-490", "659", "392", "48" ], "h": [ "0" ], "claimed_group": [ 2, 2, 2, 4 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "176400.cr.1" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_2.2.2.4.json" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-17 06:01:56 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "0.820000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.2.4.c.log", "conductor_log": "data/logs/2.2.2.4.c.conductor.log", "conductor_cputime_seconds": 0.28 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.2.6.a", "group": [ 2, 2, 2, 6 ], "group_key": "2.2.2.6", "group_bracket": "[2,2,2,6]", "order": 48, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "2", "-12", "11", "18", "-14", "-6" ], "h": [ "1", "0", "1" ], "f_str": "-6*x^5 - 14*x^4 + 18*x^3 + 11*x^2 - 12*x + 2", "h_str": "x^2 + 1", "g": [ "9", "-48", "46", "72", "-55", "-24" ], "g_str": "-24*x^5 - 55*x^4 + 72*x^3 + 46*x^2 - 48*x + 9", "g_scale": "1", "discriminant": "3613040640000", "discriminant_digits": 13, "igusa_clebsch": [ "117664", "137798464", "5401894811136", "2081111408640000" ], "g2_invariants": [ "672153012378322157/496175625", "47092767630329177/992351250", "401770096564/245025" ], "rational_weierstrass_points": 6, "minimal_model": { "f": [ "0", "6", "-14", "-18", "11", "12", "2" ], "h": [ "0", "1", "0", "1" ] }, "minimal_discriminant": "508083840000", "minimal_discriminant_factored": "2^10 * 3^8 * 5^4 * 11^2", "conductor": "39600", "conductor_factored": "2^4 * 3^2 * 5^2 * 11", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "39600.dq.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B2%2C-12%2C11%2C18%2C-14%2C-6%5D%2C%5B1%2C0%2C1%5D%5D", "snapshot_label": "39600.dq.1" }, "torsion": { "invariants": [ 2, 2, 2, 6 ], "order": 48, "generators": [ { "order": 2, "mumford": [ "x^2 - 17/24*x + 1/8", "0", "2" ] }, { "order": 2, "mumford": [ "x + 1", "0", "1" ] }, { "order": 2, "mumford": [ "x^2 + 8/3*x - 1", "0", "2" ] }, { "order": 6, "mumford": [ "x^2 - 3/8*x", "-8*x + 3", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 48, "point_count_primes": [ 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 2, 2, 2, 6 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 23, "chi": "T^4 - 4*T^3 - 2*T^2 - 92*T + 529", "primes_tried_below": 500, "admissible_primes": 5, "reducible_chi": 4 }, "q_simple": { "certified": true, "prime": 23, "chi": "T^4 - 4*T^3 - 2*T^2 - 92*T + 529" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 42, "reducible_chi": 23 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 69 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "2", "-12", "11", "18", "-14", "-6" ], "h": [ "1", "0", "1" ], "claimed_group": [ 2, 2, 2, 6 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "39600.dq.1" }, "source": { "kind": "import", "name": "paper-t1-069", "file": "table1.txt", "line": 75, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:33:04 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.360000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.2.6.a.log", "conductor_log": "data/logs/2.2.2.6.a.conductor.log", "conductor_cputime_seconds": 0.2 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.2.6.b", "group": [ 2, 2, 2, 6 ], "group_key": "2.2.2.6", "group_bracket": "[2,2,2,6]", "order": 48, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "0", "30", "39", "-72", "-111", "-30" ], "h": [ "0", "1", "1" ], "f_str": "-30*x^5 - 111*x^4 - 72*x^3 + 39*x^2 + 30*x", "h_str": "x^2 + x", "g": [ "0", "120", "157", "-286", "-443", "-120" ], "g_str": "-120*x^5 - 443*x^4 - 286*x^3 + 157*x^2 + 120*x", "g_scale": "1", "discriminant": "2870129313999360000", "discriminant_digits": 19, "igusa_clebsch": [ "1027592", "35035401604", "10489160175110792", "41329862121590784000000" ], "g2_invariants": [ "34966618702761485299117249/10090298369529000000", "253148116783272010201/3737147544270000", "-1921309006090849/4152386160300" ], "rational_weierstrass_points": 6, "minimal_model": { "f": [ "0", "30", "39", "-72", "-111", "-30" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "10090298369529000000", "minimal_discriminant_factored": "2^6 * 3^6 * 5^6 * 7^12", "conductor": "900", "conductor_factored": "30 * 30 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 30 * 30 for the elliptic curves ['1', '0', '1', '-334', '-2368'] and ['1', '0', '1', '-19', '26'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "900" }, "lmfdb": { "label": "900.b.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C30%2C39%2C-72%2C-111%2C-30%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "900.b.2" }, "torsion": { "invariants": [ 2, 2, 2, 6 ], "order": 48, "generators": [ { "order": 2, "mumford": [ "x^2 + 31/15*x - 8/5", "0", "2" ] }, { "order": 2, "mumford": [ "x - 3/5", "0", "1" ] }, { "order": 2, "mumford": [ "x^2 + x", "0", "2" ] }, { "order": 6, "mumford": [ "x^2 - 3/2*x + 5/2", "-55/2*x - 65/2", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 144, "point_count_primes": [ 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 2, 2, 2, 6 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 91, "reducible_chi": 91 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x - 3/5*z : -12/25*x^3 + 2/25*x^2*z + 38/25*x*z^2 + 343/125*y + 12/25*z^3 : -8/5*x - z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "0", "1", "-334", "-2368" ], [ "1", "0", "1", "-19", "26" ] ], "conductors": [ "30", "30" ], "conductor": "900", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 42, "reducible_chi": 42 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 74 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "30", "39", "-72", "-111", "-30" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 2, 2, 6 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "900.b.2" }, "source": { "kind": "import", "name": "paper-t2-074", "file": "table2.txt", "line": 80, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:41:14 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.15000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.2.6.b.log", "conductor_log": "data/logs/2.2.2.6.b.conductor.log", "conductor_cputime_seconds": 0.15 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.2.6.c", "group": [ 2, 2, 2, 6 ], "group_key": "2.2.2.6", "group_bracket": "[2,2,2,6]", "order": 48, "class": "gsplit", "class_certified": true, "class_text": "geometrically split, simple over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "1355999975081200", "2757085929534360", "16095094765893", "14380189111", "-25655829", "6993" ], "h": [ "0", "1", "1" ], "f_str": "6993*x^5 - 25655829*x^4 + 14380189111*x^3 + 16095094765893*x^2 + 2757085929534360*x + 1355999975081200", "h_str": "x^2 + x", "g": [ "5423999900324800", "11028343718137440", "64380379063573", "57520756446", "-102623315", "27972" ], "g_str": "27972*x^5 - 102623315*x^4 + 57520756446*x^3 + 64380379063573*x^2 + 11028343718137440*x + 5423999900324800", "g_scale": "1", "discriminant": "35878448948066852935038381285129780531100823226415767133165680799200216725449356333597696000000", "discriminant_digits": 95, "igusa_clebsch": [ "137902064479437234244616", "35349820210668815871367636586314097691372004", "1679343529592066882211565644932964124219164274706616012250849557224", "28072474696037819160080651815777591982258215701736351019498522360973570545896703527104878017265664000000" ], "g2_invariants": [ "1521962333789126326776453562642735921183791307522788286142281658116496346600156845875930488004258549143399286657/6853631517587358193379065384711326167543509692806726323119756435784563121556812384547089359684000000", "61652267902306842394735058419607160456531564587392998002891497586947944591143633123086307823986191760651/6867359368965921069597190167856872054781016945681142927832493590471916482605540860727684000000", "1893822486532904518439205315058854663384055713871982853517500985123697363611917449/4080789825673233944205310843084306972758156215128030096638733172472228096" ], "rational_weierstrass_points": 6, "minimal_model": { "f": [ "1355999975081200", "2757085929534360", "16095094765893", "14380189111", "-25655829", "6993" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "6853631517587358193379065384711326167543509692806726323119756435784563121556812384547089359684000000", "minimal_discriminant_factored": "2^8 * 3^24 * 5^6 * 7^4 * 11^6 * 17^12 * 23^6 * 31^12 * 37^6 * 67^12", "conductor": "168017780700", "conductor_factored": "2^2 * 3^2 * 5^2 * 7^2 * 11^2 * 23 * 37^2", "conductor_note": "", "conductor_source": "magma" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B1355999975081200%2C2757085929534360%2C16095094765893%2C14380189111%2C-25655829%2C6993%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 2, 2, 6 ], "order": 48, "generators": [ { "order": 2, "mumford": [ "x^2 + 1301263/3996*x + 640867/3996", "0", "2" ] }, { "order": 2, "mumford": [ "x + 73/148", "0", "1" ] }, { "order": 2, "mumford": [ "x^2 - 4263*x + 4501880", "0", "2" ] }, { "order": 6, "mumford": [ "x^2 - 526551/925*x - 208749508/925", "-29157374916/4625*x - 8023716816228/4625", "2" ] } ], "method": "generators", "generators_note": "", "generators_submitted": 4, "point_count_gcd": 48, "point_count_primes": [ 13, 19, 29, 41, 43, 47, 53, 59, 61, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149 ], "claimed": [ 2, 2, 2, 6 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 85, "reducible_chi": 59 }, "q_simple": { "certified": true, "prime": 29, "chi": "T^4 - 26*T^2 + 841" }, "split": { "geometrically_split": true, "split_over_Q": false, "certificate": { "kind": "richelot", "over": "Q", "detail": "(2,2)-isogenous over Q to a Weil restriction of an elliptic curve E over Q(sqrt(-11)) (the surface is E x E^sigma)" }, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 36, "reducible_chi": 26 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "F. Najman", "year": 2026, "historical": false }, "reference": "new", "sources": [], "rm": false, "notes": "First Q-simple, geometrically split example of [2,2,2,6] for the census (the gsplit cell is empty). Construction: quadratic-conjugate (2,2)-gluing. E = E_{2,6}(t) (Tate normal form of HLP Table 3) with t = (11 + 84*sqrt(-11))/37 over K = Q(sqrt(-11)), E(K)_tors = Z/2 x Z/6, glued with its Galois conjugate E^sigma along the G_K-isomorphism E[2] -> E^sigma[2], T_i -> sigma(T_i) (it satisfies psi^sigma psi = id, so the glued surface and the curve descend to Q). The parameter was chosen so that the Kummer classes of both generators of E(K)_tors/2 are sigma-invariant: N(t-3), N((t-5)(t+3)) and N((t-9)(t-1)) are squares in Q (norms from K); this doubles the torsion twice, J(Q)_tors = [2,2,2,6] against [2,6] for a generic member. E is not a Q-curve (a_P(E) <> a_P(E^sigma) at 13 of 32 split primes below 400), so J is simple over Q (also certified by chi_29 irreducible); J is (2,2)-isogenous over Q to Res_{K/Q} E, which Magma's Richelot search finds. Magma's Conductor on the reduced minimal model: 2^2 * 3^2 * 5^2 * 7^2 * 11^2 * 23 * 37^2. Rigorous conductor: J is (2,2)-isogenous over Q to Res_(K/Q) E with K the quadratic field of discriminant -11, so N(J) = |d_K|^2 * Norm(f_E) = 121 * 1388576700 = 168017780700 (computed with Magma; conductor is an isogeny invariant).", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "1355999975081200", "2757085929534360", "16095094765893", "14380189111", "-25655829", "6993" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 2, 2, 6 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "file", "path": "submissions/inbox/gsplit_2.2.2.6.json" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-17 06:01:57 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.30-1", "cputime_seconds": "0.810000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.2.6.c.log", "conductor_log": "data/logs/2.2.2.6.c.conductor.log", "conductor_cputime_seconds": 0.24 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.2.8.a", "group": [ 2, 2, 2, 8 ], "group_key": "2.2.2.8", "group_bracket": "[2,2,2,8]", "order": 64, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "463250390625", "463480444900", "230082726", "28452", "1" ], "h": [ "0" ], "f_str": "x^5 + 28452*x^4 + 230082726*x^3 + 463480444900*x^2 + 463250390625*x", "h_str": "0", "g": [ "0", "1853001562500", "1853921779600", "920330904", "113808", "4" ], "g_str": "4*x^5 + 113808*x^4 + 920330904*x^3 + 1853921779600*x^2 + 1853001562500*x", "g_scale": "1", "discriminant": "745490858127960076536424421686729162997813125054464000000000000000000000000", "discriminant_digits": 75, "igusa_clebsch": [ "1706492239111274496", "26202568217339292086428292874240000", "15089203650753430871003215460499623453199622471680000", "11927853730047361224582790746987666607965010000871424000000000000000000000000" ], "g2_invariants": [ "1428647890681506117517030679017448135110089546848896994343256/9420108576968268974828649861037731170654296875", "611486704319699795597016434572896516300439415245745241404753/113041302923619227697943798332452774047851562500", "49454399983335977741546963952554081830387705299793343993641/254342931578143262320373546248018741607666015625" ], "rational_weierstrass_points": 6, "minimal_model": { "f": [ "772392807", "473592636", "-27768796", "-120020", "14167", "112" ], "h": [ "0", "1" ] }, "minimal_discriminant": "9376097829696673222178250408886962890625000000000000", "minimal_discriminant_factored": "2^12 * 3^22 * 5^24 * 7^4 * 11^12 * 13^2 * 31^2", "conductor": "71089200", "conductor_factored": "2^4 * 3^2 * 5^2 * 7^2 * 13 * 31", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C463250390625%2C463480444900%2C230082726%2C28452%2C1%5D%2C%5B0%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 2, 2, 8 ], "order": 64, "generators": [ { "order": 2, "mumford": [ "x^2 + x", "0", "2" ] }, { "order": 2, "mumford": [ "x^2 + 3026*x + 3025", "0", "2" ] }, { "order": 2, "mumford": [ "x^2 + 12826*x + 29648025", "0", "2" ] }, { "order": 8, "mumford": [ "x^2 - 1559*x + 18150000", "907620*x - 3017437500", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 64, "point_count_primes": [ 17, 19, 23, 29, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131 ], "claimed": [ 2, 2, 2, 8 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 41, "chi": "T^4 + 4*T^3 + 6*T^2 + 164*T + 1681", "primes_tried_below": 500, "admissible_primes": 6, "reducible_chi": 4 }, "q_simple": { "certified": true, "prime": 29, "chi": "T^4 - 10*T^2 + 841" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 39, "reducible_chi": 24 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Balakrishnan–Najman–Shnidman–Sutherland 2026", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 70 (source: new)", "sources": [ "BNSS2026" ], "rm": false, "notes": "new (this paper)", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "463250390625", "463480444900", "230082726", "28452", "1" ], "h": [ "0" ], "claimed_group": [ 2, 2, 2, 8 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "import", "name": "paper-t1-070", "file": "table1.txt", "line": 76, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:33:04 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.380000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.2.8.a.log", "conductor_log": "data/logs/2.2.2.8.a.conductor.log", "conductor_cputime_seconds": 0.32 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.2.8.b", "group": [ 2, 2, 2, 8 ], "group_key": "2.2.2.8", "group_bracket": "[2,2,2,8]", "order": 64, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "-45", "42", "30", "-23", "-9", "3", "1" ], "h": [ "0", "1", "1" ], "f_str": "x^6 + 3*x^5 - 9*x^4 - 23*x^3 + 30*x^2 + 42*x - 45", "h_str": "x^2 + x", "g": [ "-180", "168", "121", "-90", "-35", "12", "4" ], "g_str": "4*x^6 + 12*x^5 - 35*x^4 - 90*x^3 + 121*x^2 + 168*x - 180", "g_scale": "1", "discriminant": "232286861721600", "discriminant_digits": 15, "igusa_clebsch": [ "369800", "35840452", "4412264150600", "232286861721600" ], "g2_invariants": [ "8441985278626659765625/2268426384", "19459687960893505625/126023688", "14889382628147500/1750329" ], "rational_weierstrass_points": 6, "minimal_model": { "f": [ "-45", "-42", "30", "22", "-9", "-3", "1" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "56710659600", "minimal_discriminant_factored": "2^4 * 3^10 * 5^2 * 7^4", "conductor": "4410", "conductor_factored": "210 * 21 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 210 * 21 for the elliptic curves ['1', '0', '0', '-1070', '7812'] and ['1', '0', '0', '-4', '-1'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "4410" }, "lmfdb": { "label": "4410.d.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-45%2C42%2C30%2C-23%2C-9%2C3%2C1%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "4410.d.2" }, "torsion": { "invariants": [ 2, 2, 2, 8 ], "order": 64, "generators": [ { "order": 2, "mumford": [ "x^2 + 9/2*x + 5", "0", "2" ] }, { "order": 2, "mumford": [ "x^2 + 1/2*x - 5", "0", "2" ] }, { "order": 2, "mumford": [ "x^2 + 1/2*x - 3", "0", "2" ] }, { "order": 8, "mumford": [ "x^2 + 27/20*x - 27/10", "49/200*x + 231/100", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 128, "point_count_primes": [ 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109 ], "claimed": [ 2, 2, 2, 8 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 91, "reducible_chi": 91 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x + z : x^2*z + x*z^2 + y : -z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "0", "0", "-1070", "7812" ], [ "1", "0", "0", "-4", "-1" ] ], "conductors": [ "210", "21" ], "conductor": "4410", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 42, "reducible_chi": 42 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Booker–Sutherland (extended database)", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 75 (source: \\cite{BookerSutherland})", "sources": [ "BookerSutherland" ], "rm": false, "notes": "", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "-45", "42", "30", "-23", "-9", "3", "1" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 2, 2, 8 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "4410.d.2" }, "source": { "kind": "import", "name": "paper-t2-075", "file": "table2.txt", "line": 81, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:41:15 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.970000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.2.8.b.log", "conductor_log": "data/logs/2.2.2.8.b.conductor.log", "conductor_cputime_seconds": 0.19 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.2.10.a", "group": [ 2, 2, 2, 10 ], "group_key": "2.2.2.10", "group_bracket": "[2,2,2,10]", "order": 80, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "-2275", "191", "3955", "-767", "-1680", "576" ], "h": [ "0" ], "f_str": "576*x^6 - 1680*x^5 - 767*x^4 + 3955*x^3 + 191*x^2 - 2275*x", "h_str": "0", "g": [ "0", "-9100", "764", "15820", "-3068", "-6720", "2304" ], "g_str": "2304*x^6 - 6720*x^5 - 3068*x^4 + 15820*x^3 + 764*x^2 - 9100*x", "g_scale": "1", "discriminant": "4113047824973942555364714086400000000", "discriminant_digits": 37, "igusa_clebsch": [ "3985217632", "138312391947710464", "174181611469137831024525312", "4113047824973942555364714086400000000" ], "g2_invariants": [ "29957828070448239182027991433943795715251/980627018207059515801600000000", "280636657894256459887273831194580529/256163792511231873515520000", "52573962524176802800787456908081/1161740555606493757440000" ], "rational_weierstrass_points": 6, "minimal_model": { "f": [ "0", "-2275", "191", "3955", "-767", "-1680", "576" ], "h": [] }, "minimal_discriminant": "1004162066644028944180838400000000", "minimal_discriminant_factored": "2^34 * 3^12 * 5^8 * 7^8 * 13^2 * 17^2", "conductor": "319360204800", "conductor_factored": "2^17 * 3^2 * 5^2 * 7^2 * 13 * 17", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C-2275%2C191%2C3955%2C-767%2C-1680%2C576%5D%2C%5B0%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 2, 2, 10 ], "order": 80, "generators": [ { "order": 2, "mumford": [ "x^2 - 7/12*x - 325/192", "0", "2" ] }, { "order": 2, "mumford": [ "x^2 + 1/24*x - 25/24", "0", "2" ] }, { "order": 2, "mumford": [ "x^2 - 10/3*x + 7/3", "0", "2" ] }, { "order": 10, "mumford": [ "x^2 - 103/24*x - 39/8", "-1103/3*x - 351", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 80, "point_count_primes": [ 11, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127 ], "claimed": [ 2, 2, 2, 10 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 11, "chi": "T^4 - 4*T^3 + 6*T^2 - 44*T + 121", "primes_tried_below": 500, "admissible_primes": 1, "reducible_chi": 0 }, "q_simple": { "certified": true, "prime": 11, "chi": "T^4 - 4*T^3 + 6*T^2 - 44*T + 121" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 40, "reducible_chi": 28 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Elkies 2024", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 71 (source: \\cite{Elkies2024})", "sources": [ "Elkies2024" ], "rm": false, "notes": "Elkies 2024", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "-2275", "191", "3955", "-767", "-1680", "576" ], "h": [ "0" ], "claimed_group": [ 2, 2, 2, 10 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "import", "name": "paper-t1-071", "file": "table1.txt", "line": 77, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:33:05 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.340000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.2.10.a.log", "conductor_log": "data/logs/2.2.2.10.a.conductor.log", "conductor_cputime_seconds": 0.23 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.2.12.a", "group": [ 2, 2, 2, 12 ], "group_key": "2.2.2.12", "group_bracket": "[2,2,2,12]", "order": 96, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "-57510204598294806758400", "9819234969339333136", "-603825803380840", "16672898913", "-211210", "1" ], "h": [ "0" ], "f_str": "x^6 - 211210*x^5 + 16672898913*x^4 - 603825803380840*x^3 + 9819234969339333136*x^2 - 57510204598294806758400*x", "h_str": "0", "g": [ "0", "-230040818393179227033600", "39276939877357332544", "-2415303213523360", "66691595652", "-844840", "4" ], "g_str": "4*x^6 - 844840*x^5 + 66691595652*x^4 - 2415303213523360*x^3 + 39276939877357332544*x^2 - 230040818393179227033600*x", "g_scale": "1", "discriminant": "21422318484333106630355721240379863061705485852383810348507836697331068850912336513023565688439182839940904384375432040998240256000000000000", "discriminant_digits": 140, "igusa_clebsch": [ "864976396011283739526283118592", "2205228184455757060741991349937322021171629164831451643904", "635389287296980774512694235623703701997130910167324204238490183734042326630535677345792", "21422318484333106630355721240379863061705485852383810348507836697331068850912336513023565688439182839940904384375432040998240256000000000000" ], "g2_invariants": [ "227593037998372769377425171781483147859944362601736519315735823621400290567983409243148195138633459637028968621125485314369268401/80555305668714798821273453211978701165384881364366063777249899756695218220841945965848560018846993868427264000000000000", "3452927323237503202452264714212997871557491125242433979996883783627889458386819377594109250776126248897373/30783177074610172097344433765962265984670245506671854407566437477570007604576918244378419200000000", "10195791909367690137145438677811586947871936912445671604337915490182989667659317/1815338946982303356419356325203632539389468490195188380498457776016512000" ], "rational_weierstrass_points": 6, "minimal_model": { "f": [ "-1282993930035013443975", "-397058962729817115", "1518598238654317", "301623595822", "-462983772", "-36750", "36" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "80555305668714798821273453211978701165384881364366063777249899756695218220841945965848560018846993868427264000000000000", "minimal_discriminant_factored": "2^24 * 3^10 * 5^12 * 7^12 * 11^12 * 13^12 * 17^2 * 19^12 * 23^4 * 41^2 * 73^2 * 109^4 * 193^2 * 241^4 * 263^2 * 409^2", "conductor": "4778692469215148637760180473900", "conductor_factored": "2^2 * 3^2 * 5^2 * 7^2 * 11^2 * 13^2 * 17 * 19^2 * 23^2 * 41 * 73 * 109 * 193 * 241 * 263 * 409", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C-57510204598294806758400%2C9819234969339333136%2C-603825803380840%2C16672898913%2C-211210%2C1%5D%2C%5B0%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 2, 2, 12 ], "order": 96, "generators": [ { "order": 2, "mumford": [ "x^2 - 128837*x + 4109579236", "0", "2" ] }, { "order": 2, "mumford": [ "x^2 - 14400*x", "0", "2" ] }, { "order": 2, "mumford": [ "x^2 - 85156*x + 1018886400", "0", "2" ] }, { "order": 12, "mumford": [ "x^2 - 64109*x + 768630940", "-166554960*x + 5515695625440", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 96, "point_count_primes": [ 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 79, 83, 89, 97, 101, 103, 107, 113, 127, 131, 137, 139, 149, 151, 157 ], "claimed": [ 2, 2, 2, 12 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 71, "chi": "T^4 - 8*T^3 + 46*T^2 - 568*T + 5041", "primes_tried_below": 500, "admissible_primes": 10, "reducible_chi": 8 }, "q_simple": { "certified": true, "prime": 59, "chi": "T^4 + 70*T^2 + 3481" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 33, "reducible_chi": 21 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Balakrishnan–Najman–Shnidman–Sutherland 2026", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 72 (source: new)", "sources": [ "BNSS2026" ], "rm": false, "notes": "new (this paper)", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "-57510204598294806758400", "9819234969339333136", "-603825803380840", "16672898913", "-211210", "1" ], "h": [ "0" ], "claimed_group": [ 2, 2, 2, 12 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "import", "name": "paper-t1-072", "file": "table1.txt", "line": 78, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:33:12 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.370000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.2.12.a.log", "conductor_log": "data/logs/2.2.2.12.a.conductor.log", "conductor_cputime_seconds": 5.99 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.2.16.a", "group": [ 2, 2, 2, 16 ], "group_key": "2.2.2.16", "group_bracket": "[2,2,2,16]", "order": 128, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "230262357769227286812882682545067236", "-80303936093332009982025661643142", "-406358837350345916288092715922", "249323508190639120726354536", "9716619250580389984680", "-42337414886067900" ], "h": [ "0", "1", "1" ], "f_str": "-42337414886067900*x^5 + 9716619250580389984680*x^4 + 249323508190639120726354536*x^3 - 406358837350345916288092715922*x^2 - 80303936093332009982025661643142*x + 230262357769227286812882682545067236", "h_str": "x^2 + x", "g": [ "921049431076909147251530730180268944", "-321215744373328039928102646572568", "-1625435349401383665152370863687", "997294032762556482905418146", "38866477002321559938721", "-169349659544271600" ], "g_str": "-169349659544271600*x^5 + 38866477002321559938721*x^4 + 997294032762556482905418146*x^3 - 1625435349401383665152370863687*x^2 - 321215744373328039928102646572568*x + 921049431076909147251530730180268944", "g_scale": "1", "discriminant": "1167259358554331315325784633207458504601629426484004213509953149159295500183418817147833967465954071833076924488318351440488852402979788064617278632285073807071525256801166238377301719010051034347950434858090729465511936000000", "discriminant_digits": 226, "igusa_clebsch": [ "6980547367801124017517990534538002202580969423721205128", "32730724227989292811651291094996580529780479649915517374263990048003650541458833328761538315294232583873924", "74291964311632502319892634826131354615125157251928986820882987406791066854721171779505763632576602833488426545713610403980206175865257154881698148835514749381768", "33476189711768179409760979326532645004238258692983261272745978521575471113867910706484578690678395213975155618096766770040965777522434818865441807466515333299992034944873377952330150157688479561684460260143082497829533766127750504066614315444506460160000000000" ], "g2_invariants": [ "179003709216674703976942035143781225779716824384230370078557037012112978669981842398117493407619971273187576993194982111464567897390566661943924996762958045933033007338663348554172848437183881764160215588674823203309084610640561446240910265003165266794161205353041/2892282071572809298525663208385903490860641125069181902385410040868286986199171793837248300335345091653860049407719177140965800196102284729519146798333164072950036434950877671233732458563167216013445645508942458490531760159841132320422095360000000000", "16809147299189253492880770988741335703196590107145830993665501088182152182018528980065261688942660471377964768659444842842724063336246293027562418830092115625702695911275602411278792324447241006910212428149496717838504443640397368269071493604345840543553/6589132881427599866511681203653919579682986006111365266729827295206358594022779486040369817481569138974596273867718185912312747009496738400056662101623918881875184717602298992624384330837931577260037499577516967430136407872860609052800000000", "484133214561242091723647522857875564016925288576418431953950371663897064593420340660868063493931231328572575392834750902322916431865350201665840536095228757587542599880632890264086281513497697930538550759/3484297743171715500341763232180477922756204121335183410414033076924918374862380728893241760373267243056124522424544955473860019848380172633134517799536531358836275814275104401034439187593062500" ], "rational_weierstrass_points": 6, "minimal_model": { "f": [ "230262357769227286812882682545067236", "-80303936093332009982025661643142", "-406358837350345916288092715922", "249323508190639120726354536", "9716619250580389984680", "-42337414886067900" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "8172897878849653176211176593391759034237856126216616521666498662494011502409157887325336594403905081536903227074405949717032660527938188199570753776004720043943367906463227039143103065841913955489370180698994750446663517121032837906888260606568960000000000", "minimal_discriminant_factored": "2^18 * 3^20 * 5^10 * 7^32 * 17^6 * 31^10 * 41^4 * 47^4 * 137^2 * 307^4 * 2081^2 * 2239^4 * 2729^4 * 119089^12 * 163729^12", "conductor": "1328257075878182563154683193645100", "conductor_factored": "477491580510 * 2781739260113226584010 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 477491580510 * 2781739260113226584010 for the elliptic curves ['1', '0', '0', '-562198913575863696451480', '160785133294502034126172196982094400'] and ['1', '0', '0', '-44159264041619640913098181712776142490', '103727250430271191833554094526364327863820764434279638100'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "1328257075878182563154683193645100" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B230262357769227286812882682545067236%2C-80303936093332009982025661643142%2C-406358837350345916288092715922%2C249323508190639120726354536%2C9716619250580389984680%2C-42337414886067900%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 2, 2, 16 ], "order": 128, "generators": [ { "order": 2, "mumford": [ "x + 1578364/2239", "0", "1" ] }, { "order": 2, "mumford": [ "x^2 + 24313182047/1035504*x - 7695188165473/258876", "0", "2" ] }, { "order": 2, "mumford": [ "x + 19722364/799", "0", "1" ] }, { "order": 16, "mumford": [ "x^2 - 617820584783507/399930795725*x - 481635448720173196314/399930795725", "36685050904415577795838069748/14797439441825*x + 3249321521204953541907986666568846/14797439441825", "2" ] } ], "method": "generators", "generators_note": "", "generators_submitted": 4, "point_count_gcd": 128, "point_count_primes": [ 11, 13, 19, 23, 29, 37, 43, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 139 ], "claimed": [ 2, 2, 2, 16 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 85, "reducible_chi": 85 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x + 127040438808088/3001354917*z : -4388826064328642/9008131337800076889*x^3 + 868483690974880588572295336670/9012199761229349577926071071*x^2*z - 7476002155784207232862311944576666/9012199761229349577926071071*x*z^2 + 7412961991040628919612801323841/27036599283688048733778213213*y + 8069445899552938542778023524/9008131337800076889*z^3 : 2921719/3001354917*x - z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "0", "0", "-562198913575863696451480", "160785133294502034126172196982094400" ], [ "1", "0", "0", "-44159264041619640913098181712776142490", "103727250430271191833554094526364327863820764434279638100" ] ], "conductors": [ "477491580510", "2781739260113226584010" ], "conductor": "1328257075878182563154683193645100", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 37, "reducible_chi": 37 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "F. Najman", "year": 2026, "historical": false }, "reference": "new", "sources": [], "rm": false, "notes": "New group [2,2,2,16]. The curve is the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4) of E1 = E_{2,4}(t) with torsion [2,4] (conductor 2781739260113226584010) and E2 = E_{2,8}(u) with torsion [2,8] (conductor 477491580510), in the Tate normal forms of HLP Table 3, at u = -20/31, t = -4333968559/38784504316. The gluing is the identity on the labelled 2-torsion points, and the Kummer images of the rational torsion of E1 and E2 agree completely (gain 4), which gives #J(Q)_tors = 2 * 4 * 4 * 4 = 128. The point was found by a fibre-wise 2-descent: over fixed u the gain locus is a 2-covering of y^2 = x^3 - d^2 x, here with d = 554577910 and a point of canonical height 22.83. The two elliptic quotients of the curve are Q-isomorphic to E1 and E2, and j(E1) != j(E2). Output of pipeline/magma/prepare_submission.m (Magma V2.29-4): J(Q)_tors = [2,2,2,16]; gcd of #J(F_p) over 25 good primes = 128 (sharp); J(F_11) = [2,2,2,16]; conductor N(E1) N(E2) = 1328257075878182563154683193645100 (J is isogenous over Q to E1 x E2). Further points on the same fibre give more curves with the same torsion subgroup (checked for one more, from 3P).", "submitter": { "name": "Filip Najman", "github": "F-Najman", "affiliation": "University of Zagreb" }, "submitted": { "f": [ "230262357769227286812882682545067236", "-80303936093332009982025661643142", "-406358837350345916288092715922", "249323508190639120726354536", "9716619250580389984680", "-42337414886067900" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 2, 2, 16 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "issue", "number": 9, "url": "https://github.com/Genus-2-torsion/Genus-2-torsion.github.io/issues/9" }, "dates": { "submitted": "2026-10-16", "verified": "2026-10-16 07:12:43 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.61000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.2.16.a.log", "conductor_log": "data/logs/2.2.2.16.a.conductor.log", "conductor_cputime_seconds": 0.63 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.4.4.a", "group": [ 2, 2, 4, 4 ], "group_key": "2.2.4.4", "group_bracket": "[2,2,4,4]", "order": 64, "class": "simple", "class_certified": true, "class_text": "geometrically simple", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "300512487407616", "414985109760", "173387808", "26065", "1" ], "h": [ "0" ], "f_str": "x^5 + 26065*x^4 + 173387808*x^3 + 414985109760*x^2 + 300512487407616*x", "h_str": "0", "g": [ "0", "1202049949630464", "1659940439040", "693551232", "104260", "4" ], "g_str": "4*x^5 + 104260*x^4 + 693551232*x^3 + 1659940439040*x^2 + 1202049949630464*x", "g_scale": "1", "discriminant": "1178771100445951599832313877222541743720097164562372538329889121081753600000000", "discriminant_digits": 79, "igusa_clebsch": [ "309361617604214784", "4070915358887772155001152055803904", "340413001503768171330589864036916918328140733874176", "18860337607135225597317022035560667899521554632997960613278225937308057600000000" ], "g2_invariants": [ "76151240140403423482619886729267482685600466657/4054938578206555962132640681574400000000", "130338153520447688748576885693749465323451/521468438555369851097304614400000000", "-57568528490485232568147702583/108771524205802975396687500" ], "rational_weierstrass_points": 6, "minimal_model": { "f": [ "14400", "3720", "-18600", "-3409", "3636", "252" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "4054938578206555962132640681574400000000", "minimal_discriminant_factored": "2^20 * 3^12 * 5^8 * 7^4 * 11^8 * 17^2 * 19^4 * 31^2", "conductor": "53430369300", "conductor_factored": "2^2 * 3^2 * 5^2 * 7^2 * 11^2 * 17 * 19 * 31", "conductor_note": "the exponent of 2 in the conductor was computed with Ogg's formula, for which Magma gives no correctness guarantee when v_2(disc) >= 12; odd exponents are rigorous", "conductor_source": "magma" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C300512487407616%2C414985109760%2C173387808%2C26065%2C1%5D%2C%5B0%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 2, 4, 4 ], "order": 64, "generators": [ { "order": 2, "mumford": [ "x^2 + 17424*x", "0", "2" ] }, { "order": 2, "mumford": [ "x + 3249", "0", "1" ] }, { "order": 4, "mumford": [ "x^2 + 2520*x - 590976", "31320*x + 286623360", "2" ] }, { "order": 4, "mumford": [ "x^2 + 11928*x + 17335296", "598488*x + 814758912", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 64, "point_count_primes": [ 13, 23, 29, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137 ], "claimed": [ 2, 2, 4, 4 ], "claim_matches": true }, "simplicity": { "geometrically_simple": true, "strict_prime": 37, "chi": "T^4 - 4*T^3 - 2*T^2 - 148*T + 1369", "primes_tried_below": 500, "admissible_primes": 4, "reducible_chi": 3 }, "q_simple": { "certified": true, "prime": 37, "chi": "T^4 - 4*T^3 - 2*T^2 - 148*T + 1369" }, "split": { "geometrically_split": false, "split_over_Q": false, "certificate": null, "isogenous_product": null, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": false, "primes_checked": 38, "reducible_chi": 22 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Balakrishnan–Najman–Shnidman–Sutherland 2026", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 73 (source: new)", "sources": [ "BNSS2026" ], "rm": false, "notes": "new (this paper)", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "300512487407616", "414985109760", "173387808", "26065", "1" ], "h": [ "0" ], "claimed_group": [ 2, 2, 4, 4 ], "claimed_class": "simple", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "import", "name": "paper-t1-073", "file": "table1.txt", "line": 79, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table1.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:33:12 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "0.350000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.4.4.a.log", "conductor_log": "data/logs/2.2.4.4.a.conductor.log", "conductor_cputime_seconds": 0.29 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.4.4.b", "group": [ 2, 2, 4, 4 ], "group_key": "2.2.4.4", "group_bracket": "[2,2,4,4]", "order": 64, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": false, "new_for_class": true, "accepted_because": "new for this class", "curve": { "f": [ "4913", "867", "-5657", "-671", "1000", "60" ], "h": [ "0", "1", "1" ], "f_str": "60*x^5 + 1000*x^4 - 671*x^3 - 5657*x^2 + 867*x + 4913", "h_str": "x^2 + x", "g": [ "19652", "3468", "-22627", "-2682", "4001", "240" ], "g_str": "240*x^5 + 4001*x^4 - 2682*x^3 - 22627*x^2 + 3468*x + 19652", "g_scale": "1", "discriminant": "862942160699747627186005239398400", "discriminant_digits": 33, "igusa_clebsch": [ "1524941576", "100225626558785284", "41201692788708076365870344", "49705468456305463325913901789347840000" ], "g2_invariants": [ "251661100242408474845787665489297281868257/12135124134840201007303198679040000", "1255747874987856457114258908128034553/4681760854490818289854629120000", "-657751693077450321646414463/1007610376741309032768300" ], "rational_weierstrass_points": 6, "minimal_model": { "f": [ "4913", "867", "-5657", "-671", "1000", "60" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "12135124134840201007303198679040000", "minimal_discriminant_factored": "2^16 * 3^6 * 5^4 * 7^8 * 11^2 * 17^12", "conductor": "485100", "conductor_factored": "210 * 2310 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 210 * 2310 for the elliptic curves ['1', '1', '1', '-70', '-205'] and ['1', '1', '1', '-5575730', '5065104575'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "485100" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B4913%2C867%2C-5657%2C-671%2C1000%2C60%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 2, 4, 4 ], "order": 64, "generators": [ { "order": 2, "mumford": [ "x + 2", "0", "1" ] }, { "order": 2, "mumford": [ "x^2 + 18*x + 17", "0", "2" ] }, { "order": 4, "mumford": [ "x^2 - 53/3*x - 34/3", "1594*x + 1088", "2" ] }, { "order": 4, "mumford": [ "x^2 + 93/10*x - 221/5", "3549/10*x - 3213/5", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 64, "point_count_primes": [ 13, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127 ], "claimed": [ 2, 2, 4, 4 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 89, "reducible_chi": 89 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x : 8/289*x^3 + 16/17*x^2*z + y : -1/17*x - z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "1", "1", "-70", "-205" ], [ "1", "1", "1", "-5575730", "5065104575" ] ], "conductors": [ "210", "2310" ], "conductor": "485100", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 40, "reducible_chi": 40 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Balakrishnan–Najman–Shnidman–Sutherland 2026", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 76 (source: new)", "sources": [ "BNSS2026" ], "rm": false, "notes": "new: 210.c5 x 2310.o4 gluing", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "4913", "867", "-5657", "-671", "1000", "60" ], "h": [ "0", "1", "1" ], "claimed_group": [ 2, 2, 4, 4 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "import", "name": "paper-t2-076", "file": "table2.txt", "line": 82, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:41:17 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.33000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.4.4.b.log", "conductor_log": "data/logs/2.2.4.4.b.conductor.log", "conductor_cputime_seconds": 0.26 }, "credit_differs": false }, { "schema": "genus-2-torsion/curve/1", "id": "2.2.4.8.a", "group": [ 2, 2, 4, 8 ], "group_key": "2.2.4.8", "group_bracket": "[2,2,4,8]", "order": 128, "class": "qsplit", "class_certified": true, "class_text": "split over ℚ", "status": "certified", "new_group": true, "new_for_class": true, "accepted_because": "new group", "curve": { "f": [ "0", "42938243694863192676456480983270400000000000000", "558098950720994323170545736900000000", "1821369625722709924500000", "2288161182841", "1" ], "h": [ "0" ], "f_str": "x^5 + 2288161182841*x^4 + 1821369625722709924500000*x^3 + 558098950720994323170545736900000000*x^2 + 42938243694863192676456480983270400000000000000*x", "h_str": "0", "g": [ "0", "171752974779452770705825923933081600000000000000", "2232395802883977292682182947600000000", "7285478502890839698000000", "9152644731364", "4" ], "g_str": "4*x^5 + 9152644731364*x^4 + 7285478502890839698000000*x^3 + 2232395802883977292682182947600000000*x^2 + 171752974779452770705825923933081600000000000000*x", "g_scale": "1", "discriminant": 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"210236485837672348145059692112335578709162510777552428286595980007180659292312445192707893374159501060170411077641379223671559818207849390993210697375099/147956739235894577478756422185530231453213486985519132264971203818189408652129648132236456500036306936066031776242147539254509568000000000000", "1612509152582781370298597776223876483970282594841957599891105131942992454171046411680909255572166038547411487716850189456932480031911735080460001593681/27741888606730233277266829159786918397477528809784837299682100715910514122274309024794335593756807550512380958045402663610220544000000000000", "1238057729861292402017510965407819438146087147319539859254369795253154453686857813015169210676694948864146563357947621881/396453629304415592395004061976532520343536748508311791981244088530330306089138274891639779392400080318050074624" ], "rational_weierstrass_points": 6, "minimal_model": { "f": [ "12400419254552180004153", "175916808980224671443", "666572690814667403", "438870061440610", "-280020979290", "32602500" ], "h": [ "0", "1", "1" ] }, "minimal_discriminant": "35953487634322382327337810591083846243130877337481149140388002527820026302467504496133458929508822585464045721626841852038845825024000000000000", "minimal_discriminant_factored": "2^36 * 3^40 * 5^12 * 7^4 * 11^4 * 13^2 * 17^6 * 19^12 * 23^4 * 29^12 * 31^2 * 53^2 * 3361^12", "conductor": "1584038398842900", "conductor_factored": "19291662390 * 82110 (product of the elliptic conductors)", "conductor_note": "N(E1) N(E2) = 19291662390 * 82110 for the elliptic curves ['1', '0', '0', '-165217981313574668360', '595398048987105563975142081600'] and ['1', '0', '0', '-49423080', '130545230400'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "conductor_source": "isogenous product of elliptic curves", "conductor_magma": "1584038398842900" }, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C42938243694863192676456480983270400000000000000%2C558098950720994323170545736900000000%2C1821369625722709924500000%2C2288161182841%2C1%5D%2C%5B0%5D%5D", "snapshot_label": "" }, "torsion": { "invariants": [ 2, 2, 4, 8 ], "order": 128, "generators": [ { "order": 2, "mumford": [ "x + 708122250000", "0", "1" ] }, { "order": 2, "mumford": [ "x^2 + 821085460000*x + 79991762432422500000000", "0", "2" ] }, { "order": 4, "mumford": [ "x^2 + 58630592979800/81*x - 870503872206430000000", "4223394679193792200/729*x + 65062152951579334665730000000/9", "2" ] }, { "order": 8, "mumford": [ "x^2 + 884501820672*x + 102861602802029606400000", "35657897448482736*x + 31152712096881595997683200000", "2" ] } ], "method": "TorsionSubgroup", "generators_note": "", "generators_submitted": 0, "point_count_gcd": 256, "point_count_primes": [ 37, 41, 43, 47, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157 ], "claimed": [ 2, 2, 4, 8 ], "claim_matches": true }, "simplicity": { "geometrically_simple": false, "strict_prime": 0, "chi": "", "primes_tried_below": 500, "admissible_primes": 83, "reducible_chi": 83 }, "q_simple": { "certified": false, "prime": 0, "chi": "" }, "split": { "geometrically_split": true, "split_over_Q": true, "certificate": { "kind": "involution", "over": "Q", "detail": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x + 708122250000*z : 20867285562291/925975574812709*y : -1/769516682841*x - z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q" }, "isogenous_product": { "ainvs": [ [ "1", "0", "0", "-165217981313574668360", "595398048987105563975142081600" ], [ "1", "0", "0", "-49423080", "130545230400" ] ], "conductors": [ "19291662390", "82110" ], "conductor": "1584038398842900", "note": "J is isogenous over Q to the product of these two elliptic curves (found by RichelotIsogenousSurfaces); the conductor is an isogeny invariant, so N(J) = N(E1) N(E2), with N(E1), N(E2) computed by Tate's algorithm" }, "cover_submitted": false, "cover_accepted": false, "cover_error": "", "signature": { "all_good_primes_fail_strictness": true, "primes_checked": 34, "reducible_chi": 34 } }, "geometrically_isomorphic_to": [], "discovery": { "by": "Balakrishnan–Najman–Shnidman–Sutherland 2026", "year": null, "historical": false }, "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 77 (source: \\cite{HLP2000})", "sources": [ "HLP2000" ], "rm": false, "notes": "new: HLP sec 3.7 cond (5) at (1/6,4/17), square-branch model", "submitter": { "name": "seed import from the Genus2Torsion repository", "github": "", "affiliation": "" }, "submitted": { "f": [ "0", "42938243694863192676456480983270400000000000000", "558098950720994323170545736900000000", "1821369625722709924500000", "2288161182841", "1" ], "h": [ "0" ], "claimed_group": [ 2, 2, 4, 8 ], "claimed_class": "split", "cover": null, "lmfdb_label_given": "" }, "source": { "kind": "import", "name": "paper-t2-077", "file": "table2.txt", "line": 83, "url": "https://github.com/AndrewVSutherland/Genus2Torsion/blob/508852a150e17c00b84b209fee23300df897871d/table2.txt" }, "dates": { "submitted": "2026-08-30", "verified": "2026-09-18 17:41:19 UTC" }, "verification": { "host": "Mordell (Dept. of Mathematics, Univ. of Zagreb)", "magma_version": "2.29-4", "cputime_seconds": "1.43000", "verify_lib_sha256": "96637e64a8daa542581fdf57bbeee1cf9396f1ec9facf171be52daef32912ee9", "log": "data/logs/2.2.4.8.a.log", "conductor_log": "data/logs/2.2.4.8.a.conductor.log", "conductor_cputime_seconds": 0.82 }, "credit_differs": true } ] }