{ "generated": "2026-11-11 19:02 UTC", "conventions": { "group": "J(Q)_tors as a list of invariant factors [n1, ..., nr], n1 | ... | nr; [] is the trivial group; key = factors joined by '.', '1' for the trivial group", "classes": { "simple": "geometrically simple", "qsplit": "split over ℚ", "gsplit": "geometrically split, simple over ℚ" }, "undecided": { "split_undecided_over_Q": "geometrically split; whether it splits over ℚ is undecided", "undecided": "neither a simplicity nor a splitness certificate was found" }, "infinite": { "grades": { "exact": "infinitely many, with this exact torsion group (certified)", "family": "a positive-dimensional family with torsion containing the group is proven; exactness for infinitely many members is open", "open": "no positive-dimensional family is recorded (this is not a claim that none exists)", "impossible": "proved impossible: no Jacobian in this class has the torsion group" }, "labels": { "exact": "∞", "family": "∞ ⊇", "open": "?", "impossible": "∅" } }, "exclusions": "proved impossibility, indexed by class; each entry has a reason, sources and a proof_url; an absent entry makes no impossibility claim", "order": "groups are listed as in the paper's tables: by number of invariant factors, then lexicographically" }, "n_groups": 96, "n_curves": 189, "n_certified": 188, "n_verified": 1, "n_groups_by_class": { "simple": 78, "qsplit": 75, "gsplit": 35, "undecided": 1 }, "n_groups_any_split": 78, "n_infinite": { "simple": { "exact": 60, "family": 0, "open": 36, "impossible": 0 }, "split": { "exact": 70, "family": 0, "open": 13, "impossible": 13 } }, "n_rejected": 0, "groups": [ { "key": "1", "group": [], "bracket": "[]", "label": "trivial", "order": 1, "rank": 0, "classes": { "simple": [ { "id": "1.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "-14580", "51520", "-63684", "31054", "-3823", "-637", "-18" ], "h": [ "1" ], "equation": { "f": "-18*x^6 - 637*x^5 - 3823*x^4 + 31054*x^3 - 63684*x^2 + 51520*x - 14580", "h": "1", "text": "y^2 + y = -18*x^6 - 637*x^5 - 3823*x^4 + 31054*x^3 - 63684*x^2 + 51520*x - 14580" }, "conductor": "277", "conductor_sort": 277, "conductor_note": "", "discriminant_digits": 31, "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" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 1 (source: \\cite{BSSVY})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 7 is strict (χ = T^4 - T^3 + 3*T^2 - 7*T + 49)", "verified": "2026-09-18 17:32:10 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "1.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "-6", "-33", "-14", "95", "-50", "3", "-1" ], "h": [ "1" ], "equation": { "f": "-x^6 + 3*x^5 - 50*x^4 + 95*x^3 - 14*x^2 - 33*x - 6", "h": "1", "text": "y^2 + y = -x^6 + 3*x^5 - 50*x^4 + 95*x^3 - 14*x^2 - 33*x - 6" }, "conductor": "1083", "conductor_sort": 1083, "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)", "discriminant_digits": 10, "lmfdb": { "label": "1083.b.390963.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/1083/b/390963/1", "snapshot_label": "" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 1 (source: \\cite{BSSVY})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:33:14 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [ { "id": "1.c", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "-7", "-37", "-122", "-129", "-42", "-5", "-7" ], "h": [ "0" ], "equation": { "f": "-7*x^6 - 5*x^5 - 42*x^4 - 129*x^3 - 122*x^2 - 37*x - 7", "h": "0", "text": "y^2 = -7*x^6 - 5*x^5 - 42*x^4 - 129*x^3 - 122*x^2 - 37*x - 7" }, "conductor": "1521", "conductor_sort": 1521, "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)", "discriminant_digits": 24, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": 2026, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "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)", "new_group": false, "new_for_class": true, "certificate": "the geometric automorphism group has order 12 (7 involutions), so the curve is bielliptic over Qbar; simple over ℚ: χ_5 = T^4 + 2*T^2 + 25 is irreducible", "verified": "2026-10-17 06:01:47 UTC", "submitted": "2026-10-16", "source_kind": "file" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": 0, "undecided": null }, "first_source": { "simple": "BSSVY", "qsplit": "BSSVY", "gsplit": "BookerSutherland", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "ThisSite" ], "note": "y² = x⁵ + x + t: every 3- and 17-integral t ≡ 3 (mod 51) has #J(𝔽₃) = 12 and #J(𝔽₁₇) = 205 (gcd 1), strict at 17 (family_1_certificate.log); the fibers t = 3, 4, 5, 7, 8 have trivial torsion, strict primes 7, 7, 7, 13, 13 and distinct G2-invariants (trivial_torsion_family.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings, in the normal form of Bröker–Howe–Lauter–Stevenhagen (arXiv:1403.6911), of E = 37a1 (torsion trivial) with all partners F anti-symplectically 3-congruent to E. These partners form a rational curve (the twist X_E^−(3) of X(3)); its parameter w gives C_w over ℚ(w) with explicit degree-3 maps onto E and F_w, so J is isogenous over ℚ to E × F_w. Every 5- and 11-integral w ≡ 2 (mod 55) has the reductions of w = 2 at 5 and 11, where J(𝔽₅) ≅ ℤ/40 and J(𝔽₁₁) ≅ ℤ/187 have coprime orders, so J(ℚ)_tors is trivial; the G2-invariants at w = 2 and w = −53 differ (family_1_split_certificate.log)", "inherited_from": null } } }, { "key": "2", "group": [ 2 ], "bracket": "[2]", "label": "ℤ/2", "order": 2, "rank": 1, "classes": { "simple": [ { "id": "2.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "-608", "389", "22", "-40", "0", "1" ], "h": [ "1", "1", "1" ], "equation": { "f": "x^5 - 40*x^3 + 22*x^2 + 389*x - 608", "h": "x^2 + x + 1", "text": "y^2 + (x^2 + x + 1)*y = x^5 - 40*x^3 + 22*x^2 + 389*x - 608" }, "conductor": "295", "conductor_sort": 295, "conductor_note": "", "discriminant_digits": 5, "lmfdb": { "label": "295.a.295.2", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/295/a/295/2", "snapshot_label": "" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 2 (source: \\cite{BSSVY})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 7 is strict (χ = T^4 - T^3 - 7*T + 49)", "verified": "2026-09-18 17:32:11 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "2.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "-56", "0", "-75", "0", "15", "0", "-1" ], "h": [ "0", "1", "0", "1" ], "equation": { "f": "-x^6 + 15*x^4 - 75*x^2 - 56", "h": "x^3 + x", "text": "y^2 + (x^3 + x)*y = -x^6 + 15*x^4 - 75*x^2 - 56" }, "conductor": "336", "conductor_sort": 336, "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)", "discriminant_digits": 9, "lmfdb": { "label": "336.a.172032.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/336/a/172032/1", "snapshot_label": "" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 2 (source: \\cite{BSSVY})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:33:15 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [ { "id": "2.c", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "-168", "-666", "-504", "-1", "-504", "665", "-168" ], "h": [ "1", "1", "1", "1" ], "equation": { "f": "-168*x^6 + 665*x^5 - 504*x^4 - x^3 - 504*x^2 - 666*x - 168", "h": "x^3 + x^2 + x + 1", "text": "y^2 + (x^3 + x^2 + x + 1)*y = -168*x^6 + 665*x^5 - 504*x^4 - x^3 - 504*x^2 - 666*x - 168" }, "conductor": "576", "conductor_sort": 576, "conductor_note": "", "discriminant_digits": 34, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": 2026, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "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)", "new_group": false, "new_for_class": true, "certificate": "(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); simple over ℚ: χ_19 = T^4 - 22*T^2 + 361 is irreducible", "verified": "2026-10-17 06:12:15 UTC", "submitted": "2026-10-16", "source_kind": "file" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": 0, "undecided": null }, "first_source": { "simple": "BSSVY", "qsplit": "BSSVY", "gsplit": "BookerSutherland", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "ThisSite", "Audit2026" ], "note": "y² = x(x⁴ + x + t): the class [(0,0) − ∞] is a rational 2-torsion point for every t (checked over ℚ(t)). Every 3- and 5-integral t ≡ 2 (mod 15) has the reductions of t = 2 at 3 and 5: #J(𝔽₃) = 22 and #J(𝔽₅) = 20 (gcd 2), strict at 3 and 5, so its torsion is exactly [2] and its Jacobian is geometrically simple (family_2_certificate.log). The record's family y² = x(x⁵ + x + t) has no rational 2-torsion generically (x⁵ + x + t is irreducible over ℚ(t)); its fiber t = 2 is exact [2] only by an accidental extra root — found by the audit of 2026-09-18", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings, in the normal form of Bröker–Howe–Lauter–Stevenhagen (arXiv:1403.6911), of E = 46a1 (torsion ℤ/2) with all partners F anti-symplectically 3-congruent to E. These partners form a rational curve (the twist X_E^−(3) of X(3)); its parameter w gives C_w over ℚ(w) with explicit degree-3 maps onto E and F_w, so J is isogenous over ℚ to E × F_w. The map onto E has prime degree, so pullback is injective on E and [2] ⊂ J(ℚ). Every 7- and 11-integral w ≡ 0 (mod 77) has the reductions of w = 0 at 7 and 11, where J(𝔽₇) ≅ ℤ/108 and J(𝔽₁₁) ≅ ℤ/70; the largest group embedding in both is [2], so J(ℚ)_tors is exactly [2]; the G2-invariants at w = 0 and w = 77 differ (family_2_split_certificate.log)", "inherited_from": null } } }, { "key": "3", "group": [ 3 ], "bracket": "[3]", "label": "ℤ/3", "order": 3, "rank": 1, "classes": { "simple": [ { "id": "3.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "0", "-1", "2", "-2", "1" ], "h": [ "1" ], "equation": { "f": "x^5 - 2*x^4 + 2*x^3 - x^2", "h": "1", "text": "y^2 + y = x^5 - 2*x^4 + 2*x^3 - x^2" }, "conductor": "997", "conductor_sort": 997, "conductor_note": "", "discriminant_digits": 6, "lmfdb": { "label": "997.b.997.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/997/b/997/1", "snapshot_label": "" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 3 (source: \\cite{BSSVY})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 3 is strict (χ = T^4 + 3*T^3 + 5*T^2 + 9*T + 9)", "verified": "2026-09-18 17:32:11 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "3.c", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "-12", "-15", "9", "31", "9", "-15", "-12" ], "h": [ "1", "1", "1", "1" ], "equation": { "f": "-12*x^6 - 15*x^5 + 9*x^4 + 31*x^3 + 9*x^2 - 15*x - 12", "h": "x^3 + x^2 + x + 1", "text": "y^2 + (x^3 + x^2 + x + 1)*y = -12*x^6 - 15*x^5 + 9*x^4 + 31*x^3 + 9*x^2 - 15*x - 12" }, "conductor": "847", "conductor_sort": 847, "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)", "discriminant_digits": 7, "lmfdb": { "label": "847.d.847.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/847/d/847/1", "snapshot_label": "" }, "sources": [], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": null, "historical": false, "first_source": null, "first_source_year": null, "torsion_method": "TorsionSubgroup", "reference": "LMFDB 847.d.847.1, https://www.lmfdb.org/Genus2Curve/Q/847/d/847/1", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-10-14 10:49:07 UTC", "submitted": "2026-10-14", "source_kind": "issue" } ], "gsplit": [ { "id": "3.b", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "-54", "-318", "-479", "92", "302", "-5", "-54" ], "h": [ "1", "1", "0", "1" ], "equation": { "f": "-54*x^6 - 5*x^5 + 302*x^4 + 92*x^3 - 479*x^2 - 318*x - 54", "h": "x^3 + x + 1", "text": "y^2 + (x^3 + x + 1)*y = -54*x^6 - 5*x^5 + 302*x^4 + 92*x^3 - 479*x^2 - 318*x - 54" }, "conductor": "324", "conductor_sort": 324, "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", "discriminant_digits": 28, "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" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 3 (source: \\cite{BSSVY})", "new_group": false, "new_for_class": true, "certificate": "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]; simple over ℚ: χ_7 = T^4 + 2*T^3 - 3*T^2 + 14*T + 49 is irreducible", "verified": "2026-09-18 17:33:17 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "BSSVY", "qsplit": null, "gsplit": "BSSVY", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite" ], "note": "y² = (x³+x+1)² − t(x²+1)³ (contact form f = h² − λQ³): D = [x² + 1, 1] has order 3 over ℚ(t). Every 3-integral t ≡ 2 (mod 3) has #J(𝔽₃) = 3, strict (family_3_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings, in the normal form of Bröker–Howe–Lauter–Stevenhagen (arXiv:1403.6911), of E = 19a3 (torsion ℤ/3) with all partners F anti-symplectically 3-congruent to E. These partners form a rational curve (the twist X_E^−(3) of X(3)); its parameter w gives C_w over ℚ(w) with explicit degree-3 maps onto E and F_w, so J is isogenous over ℚ to E × F_w. The map onto E has prime degree, so pullback is injective on E and [3] ⊂ J(ℚ); F_w also inherits a rational point of order 3 from E, but no extra 3-torsion appears in J on the class below. Every 5- and 7-integral w ≡ −1 (mod 35) has the reductions of w = −1 at 5 and 7, where J(𝔽₅) ≅ [3,3] and J(𝔽₇) ≅ ℤ/54; the largest group embedding in both is [3], so J(ℚ)_tors is exactly [3]; the G2-invariants at w = −1 and w = 34 differ (family_3_split_certificate.log)", "inherited_from": null } } }, { "key": "4", "group": [ 4 ], "bracket": "[4]", "label": "ℤ/4", "order": 4, "rank": 1, "classes": { "simple": [ { "id": "4.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "-1", "0", "1" ], "h": [ "1", "0", "0", "1" ], "equation": { "f": "x^3 - x", "h": "x^3 + 1", "text": "y^2 + (x^3 + 1)*y = x^3 - x" }, "conductor": "1070", "conductor_sort": 1070, "conductor_note": "", "discriminant_digits": 7, "lmfdb": { "label": "1070.a.2140.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/1070/a/2140/1", "snapshot_label": "" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 4 (source: \\cite{BSSVY})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 17 is strict (χ = T^4 + 4*T^3 + 6*T^2 + 68*T + 289)", "verified": "2026-09-18 17:32:12 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "4.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "-7536", "-4824", "-4239", "-744", "-304", "76", "-4" ], "h": [ "0", "0", "1", "1" ], "equation": { "f": "-4*x^6 + 76*x^5 - 304*x^4 - 744*x^3 - 4239*x^2 - 4824*x - 7536", "h": "x^3 + x^2", "text": "y^2 + (x^3 + x^2)*y = -4*x^6 + 76*x^5 - 304*x^4 - 744*x^3 - 4239*x^2 - 4824*x - 7536" }, "conductor": "289", "conductor_sort": 289, "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", "discriminant_digits": 34, "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" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 4 (source: \\cite{BSSVY})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:35:49 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [ { "id": "4.c", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "-9", "-44", "11", "80", "105", "56", "16" ], "h": [ "0", "1" ], "equation": { "f": "16*x^6 + 56*x^5 + 105*x^4 + 80*x^3 + 11*x^2 - 44*x - 9", "h": "x", "text": "y^2 + x*y = 16*x^6 + 56*x^5 + 105*x^4 + 80*x^3 + 11*x^2 - 44*x - 9" }, "conductor": "775", "conductor_sort": 775, "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", "discriminant_digits": 20, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": 2026, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "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)", "new_group": false, "new_for_class": true, "certificate": "(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); simple over ℚ: χ_3 = T^4 - 2*T^2 + 9 is irreducible", "verified": "2026-10-17 06:28:59 UTC", "submitted": "2026-10-16", "source_kind": "file" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": 0, "undecided": null }, "first_source": { "simple": "BSSVY", "qsplit": "BSSVY", "gsplit": "BookerSutherland", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite" ], "note": "y² = x·Q₄(x), Q₄ the norm from ℚ(√2) of the automatic-halving quadratic q_{α,β}, α = 1 + s√2, β = 1: Q₄ = u² + xl² with u = x² + x + 1 − 2s², l = 2x + 2, so D = [u, xl mod u] has 2D = [x, 0] ≠ 0 and order 4 over ℚ(s). Every s ≡ 1 (mod 57) has #J(𝔽₃) = 4 and #J(𝔽₁₉) = 412, strict at 19 (family_4_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings, in the normal form of Bröker–Howe–Lauter–Stevenhagen (arXiv:1403.6911), of E = 17a4 (torsion ℤ/4) with all partners F anti-symplectically 3-congruent to E. These partners form a rational curve (the twist X_E^−(3) of X(3)); its parameter w gives C_w over ℚ(w) with explicit degree-3 maps onto E and F_w, so J is isogenous over ℚ to E × F_w. The map onto E has prime degree, so pullback is injective on E and [4] ⊂ J(ℚ). Every 5- and 7-integral w ≡ 2 (mod 35) has the reductions of w = 2 at 5 and 7, where J(𝔽₅) ≅ ℤ/40 and J(𝔽₇) ≅ ℤ/28; the largest group embedding in both is [4], so J(ℚ)_tors is exactly [4]; the G2-invariants at w = 2 and w = −33 differ (family_4_split_certificate.log)", "inherited_from": null } } }, { "key": "5", "group": [ 5 ], "bracket": "[5]", "label": "ℤ/5", "order": 5, "rank": 1, "classes": { "simple": [ { "id": "5.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "-6", "11", "-19", "14", "-9", "1" ], "h": [ "1" ], "equation": { "f": "x^5 - 9*x^4 + 14*x^3 - 19*x^2 + 11*x - 6", "h": "1", "text": "y^2 + y = x^5 - 9*x^4 + 14*x^3 - 19*x^2 + 11*x - 6" }, "conductor": "277", "conductor_sort": 277, "conductor_note": "", "discriminant_digits": 5, "lmfdb": { "label": "277.a.277.2", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/277/a/277/2", "snapshot_label": "" }, "sources": [ "Ogg1973" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": true, "year": null, "historical": false, "first_source": "Ogg1973", "first_source_year": 1973, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 5 (source: \\cite{Ogg1973})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 3 is strict (χ = T^4 + T^3 + T^2 + 3*T + 9)", "verified": "2026-09-18 17:32:12 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "5.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "-344", "348", "-334", "17", "14", "-36", "-8" ], "h": [ "0" ], "equation": { "f": "-8*x^6 - 36*x^5 + 14*x^4 + 17*x^3 - 334*x^2 + 348*x - 344", "h": "0", "text": "y^2 = -8*x^6 - 36*x^5 + 14*x^4 + 17*x^3 - 334*x^2 + 348*x - 344" }, "conductor": "121", "conductor_sort": 121, "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", "discriminant_digits": 24, "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" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 5 (source: \\cite{BSSVY})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:35:50 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [ { "id": "5.c", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "357", "30", "86", "-21", "2", "-3" ], "h": [ "1", "0", "0", "1" ], "equation": { "f": "-3*x^5 + 2*x^4 - 21*x^3 + 86*x^2 + 30*x + 357", "h": "x^3 + 1", "text": "y^2 + (x^3 + 1)*y = -3*x^5 + 2*x^4 - 21*x^3 + 86*x^2 + 30*x + 357" }, "conductor": "676", "conductor_sort": 676, "conductor_note": "", "discriminant_digits": 21, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": 2026, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "generators", "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)", "new_group": false, "new_for_class": true, "certificate": "(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); simple over ℚ: χ_5 = T^4 - T^2 + 25 is irreducible", "verified": "2026-10-17 06:29:00 UTC", "submitted": "2026-10-16", "source_kind": "file" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": 0, "undecided": null }, "first_source": { "simple": "Ogg1973", "qsplit": "BSSVY", "gsplit": "BookerSutherland", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "ThisSite", "Najman2026" ], "note": "the contact-5 family y² = (1 + x + x²)² − tx⁵: [x, 1] = [(0,1) − ∞] has order 5 over ℚ(t). Every 3-integral t ≡ 1 (mod 3) has #J(𝔽₃) = 5, strict (family_5_certificate.log). The record names no family for [5]; the normalized five-torsion chart f = h² − (1+a+b)²x⁵, h = 1 + ax + bx², of the lab inventory it refers to has f(1) = 0 for all a, b, so its torsion contains [10] (audit of 2026-10-18)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings, in the normal form of Bröker–Howe–Lauter–Stevenhagen (arXiv:1403.6911), of E = 11a3 (torsion ℤ/5) with all partners F anti-symplectically 3-congruent to E. These partners form a rational curve (the twist X_E^−(3) of X(3)); its parameter w gives C_w over ℚ(w) with explicit degree-3 maps onto E and F_w, so J is isogenous over ℚ to E × F_w. The map onto E has prime degree, so pullback is injective on E and [5] ⊂ J(ℚ). Every 5- and 7-integral w ≡ 10 (mod 35) has the reductions of w = 10 at 5 and 7, where J(𝔽₅) ≅ [5,5] and J(𝔽₇) ≅ ℤ/70; the largest group embedding in both is [5], so J(ℚ)_tors is exactly [5]; the G2-invariants at w = 10 and w = −25 differ (family_5_split_certificate.log)", "inherited_from": null } } }, { "key": "6", "group": [ 6 ], "bracket": "[6]", "label": "ℤ/6", "order": 6, "rank": 1, "classes": { "simple": [ { "id": "6.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "3", "21", "46", "26", "-12", "1" ], "h": [ "0", "1", "1" ], "equation": { "f": "x^5 - 12*x^4 + 26*x^3 + 46*x^2 + 21*x + 3", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = x^5 - 12*x^4 + 26*x^3 + 46*x^2 + 21*x + 3" }, "conductor": "1038", "conductor_sort": 1038, "conductor_note": "", "discriminant_digits": 6, "lmfdb": { "label": "1038.a.1038.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/1038/a/1038/1", "snapshot_label": "" }, "sources": [ "Flynn1991" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": true, "year": null, "historical": false, "first_source": "Flynn1991", "first_source_year": 1991, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 6 (source: \\cite{Flynn1991})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 17 is strict (χ = T^4 - 4*T^3 + 10*T^2 - 68*T + 289)", "verified": "2026-09-18 17:32:13 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "6.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "-92", "42", "3", "37", "-12", "-3", "-5" ], "h": [ "0", "1", "1" ], "equation": { "f": "-5*x^6 - 3*x^5 - 12*x^4 + 37*x^3 + 3*x^2 + 42*x - 92", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = -5*x^6 - 3*x^5 - 12*x^4 + 37*x^3 + 3*x^2 + 42*x - 92" }, "conductor": "196", "conductor_sort": 196, "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)", "discriminant_digits": 16, "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" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 6 (source: \\cite{BSSVY})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:35:52 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [ { "id": "6.c", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "-7", "27", "-18", "-23", "6", "3" ], "h": [ "1", "0", "0", "1" ], "equation": { "f": "3*x^5 + 6*x^4 - 23*x^3 - 18*x^2 + 27*x - 7", "h": "x^3 + 1", "text": "y^2 + (x^3 + 1)*y = 3*x^5 + 6*x^4 - 23*x^3 - 18*x^2 + 27*x - 7" }, "conductor": "657", "conductor_sort": 657, "conductor_note": "", "discriminant_digits": 13, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": 2026, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "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)", "new_group": false, "new_for_class": true, "certificate": "(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); simple over ℚ: χ_5 = T^4 - 2*T^2 + 25 is irreducible", "verified": "2026-10-17 06:29:01 UTC", "submitted": "2026-10-16", "source_kind": "file" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": 0, "undecided": null }, "first_source": { "simple": "Flynn1991", "qsplit": "BSSVY", "gsplit": "BookerSutherland", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "ThisSite", "Najman2026" ], "note": "the contact-6 family h = t + x + x² + x³, y² = h² − (x−1)⁶: [(1, h(1)) − ∞] has order 6 over ℚ(t). Every t ≡ 16 (mod 33) has #J(𝔽₃) = 12 and #J(𝔽₁₁) = 102 (gcd 6), strict at 11 (family_6_certificate.log). The record's family, the plain M(12) chart T₁₂ = ax² − x + r, h = (x−r)(T₁₂+1), f = ax²T₁₂(T₁₂+1), has torsion exactly [6] generically, but its class of order 2 is divisible by 2 in J(𝔽_p) in every good residue class tested, so it has no explicit residue-class certificate (audit of 2026-10-18)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings, in the normal form of Bröker–Howe–Lauter–Stevenhagen (arXiv:1403.6911), of E = 14a4 (torsion ℤ/6) with all partners F anti-symplectically 3-congruent to E. These partners form a rational curve (the twist X_E^−(3) of X(3)); its parameter w gives C_w over ℚ(w) with explicit degree-3 maps onto E and F_w, so J is isogenous over ℚ to E × F_w. The map onto E has prime degree, so pullback is injective on E and [6] ⊂ J(ℚ); F_w also inherits a rational point of order 3 from E, but no extra 3-torsion appears in J on the class below. Every 11- and 17-integral w ≡ −9 (mod 187) has the reductions of w = −9 at 11 and 17, where J(𝔽₁₁) ≅ [2,54] and J(𝔽₁₇) ≅ [3,84]; the largest group embedding in both is [6], so J(ℚ)_tors is exactly [6]; the G2-invariants at w = −9 and w = 178 differ (family_6_split_certificate.log)", "inherited_from": null } } }, { "key": "7", "group": [ 7 ], "bracket": "[7]", "label": "ℤ/7", "order": 7, "rank": 1, "classes": { "simple": [ { "id": "7.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "-2", "3", "0", "-3", "0", "1" ], "h": [ "0", "0", "0", "1" ], "equation": { "f": "x^5 - 3*x^3 + 3*x - 2", "h": "x^3", "text": "y^2 + x^3*y = x^5 - 3*x^3 + 3*x - 2" }, "conductor": "461", "conductor_sort": 461, "conductor_note": "", "discriminant_digits": 7, "lmfdb": { "label": "461.a.461.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/461/a/461/1", "snapshot_label": "" }, "sources": [ "Ogg1973" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": true, "year": null, "historical": false, "first_source": "Ogg1973", "first_source_year": 1973, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 7 (source: \\cite{Ogg1973})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 5 is strict (χ = T^4 - T^3 + T^2 - 5*T + 25)", "verified": "2026-09-18 17:32:14 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "7.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "-3", "-4", "2", "7", "2", "-4", "-3" ], "h": [ "1", "0", "0", "1" ], "equation": { "f": "-3*x^6 - 4*x^5 + 2*x^4 + 7*x^3 + 2*x^2 - 4*x - 3", "h": "x^3 + 1", "text": "y^2 + (x^3 + 1)*y = -3*x^6 - 4*x^5 + 2*x^4 + 7*x^3 + 2*x^2 - 4*x - 3" }, "conductor": "676", "conductor_sort": 676, "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)", "discriminant_digits": 11, "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" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 7 (source: \\cite{BSSVY})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:35:53 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [ { "id": "7.c", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "2", "2", "-8", "-6", "12", "10", "2" ], "h": [ "0" ], "equation": { "f": "2*x^6 + 10*x^5 + 12*x^4 - 6*x^3 - 8*x^2 + 2*x + 2", "h": "0", "text": "y^2 = 2*x^6 + 10*x^5 + 12*x^4 - 6*x^3 - 8*x^2 + 2*x + 2" }, "conductor": "1521", "conductor_sort": 1521, "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)", "discriminant_digits": 14, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": 2026, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "generators", "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)", "new_group": false, "new_for_class": true, "certificate": "the geometric automorphism group has order 12 (7 involutions), so the curve is bielliptic over Qbar; simple over ℚ: χ_5 = T^4 + 2*T^2 + 25 is irreducible", "verified": "2026-10-17 06:29:02 UTC", "submitted": "2026-10-16", "source_kind": "file" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": 0, "undecided": null }, "first_source": { "simple": "Ogg1973", "qsplit": "BSSVY", "gsplit": "BookerSutherland", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite" ], "note": "contact-7 chart h = 1 − (7/2)x + ax² + bx³, f = (h² + (x−1)⁷)/x², slice a = t, b = 1: [(1, h(1)) − ∞] has order 7 over ℚ(t). Every 3-integral t ≡ 2 (mod 3) has #J(𝔽₃) = 7, strict (family_7_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings, in the normal form of Bröker–Howe–Lauter–Stevenhagen (arXiv:1403.6911), of E = 26b1 (torsion ℤ/7) with all partners F anti-symplectically 3-congruent to E. These partners form a rational curve (the twist X_E^−(3) of X(3)); its parameter w gives C_w over ℚ(w) with explicit degree-3 maps onto E and F_w, so J is isogenous over ℚ to E × F_w. The map onto E has prime degree, so pullback is injective on E and [7] ⊂ J(ℚ). Every 7- and 11-integral w ≡ 0 (mod 77) has the reductions of w = 0 at 7 and 11, where J(𝔽₇) ≅ ℤ/91 and J(𝔽₁₁) ≅ ℤ/238; the largest group embedding in both is [7], so J(ℚ)_tors is exactly [7]; the G2-invariants at w = 0 and w = 77 differ (family_7_split_certificate.log)", "inherited_from": null } } }, { "key": "8", "group": [ 8 ], "bracket": "[8]", "label": "ℤ/8", "order": 8, "rank": 1, "classes": { "simple": [ { "id": "8.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "0", "0", "-1", "-2", "-2", "-1" ], "h": [ "1", "1" ], "equation": { "f": "-x^6 - 2*x^5 - 2*x^4 - x^3", "h": "x + 1", "text": "y^2 + (x + 1)*y = -x^6 - 2*x^5 - 2*x^4 - x^3" }, "conductor": "464", "conductor_sort": 464, "conductor_note": "", "discriminant_digits": 7, "lmfdb": { "label": "464.a.464.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/464/a/464/1", "snapshot_label": "" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 8 (source: \\cite{BSSVY})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 3 is strict (χ = T^4 + T^3 + 2*T^2 + 3*T + 9)", "verified": "2026-09-18 17:32:14 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "8.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "-15", "0", "7", "0", "-4" ], "h": [ "0", "1", "0", "1" ], "equation": { "f": "-4*x^4 + 7*x^2 - 15", "h": "x^3 + x", "text": "y^2 + (x^3 + x)*y = -4*x^4 + 7*x^2 - 15" }, "conductor": "300", "conductor_sort": 300, "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)", "discriminant_digits": 15, "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" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 8 (source: \\cite{BSSVY})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:35:54 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [ { "id": "8.c", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "-1", "-2", "1", "4", "-1", "-2", "1" ], "h": [ "0" ], "equation": { "f": "x^6 - 2*x^5 - x^4 + 4*x^3 + x^2 - 2*x - 1", "h": "0", "text": "y^2 = x^6 - 2*x^5 - x^4 + 4*x^3 + x^2 - 2*x - 1" }, "conductor": "2560", "conductor_sort": 2560, "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", "discriminant_digits": 11, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": 2026, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "generators", "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)", "new_group": false, "new_for_class": true, "certificate": "(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); simple over ℚ: χ_3 = T^4 - 2*T^2 + 9 is irreducible", "verified": "2026-10-17 06:29:03 UTC", "submitted": "2026-10-16", "source_kind": "file" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": 0, "undecided": null }, "first_source": { "simple": "BSSVY", "qsplit": "BSSVY", "gsplit": "BookerSutherland", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite" ], "note": "the A(8) chart, slice p = 2, r = 3 with free parameter t: a class D₈ of order 8 over ℚ(t). Every t ≡ 3 (mod 95) has #J(𝔽₅) = 16 and #J(𝔽₁₉) = 584 (gcd 8), strict at 19 (family_8_certificate.log); the class contains the record's fiber t = 3", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings, in the normal form of Bröker–Howe–Lauter–Stevenhagen (arXiv:1403.6911), of E = 15a4 (torsion ℤ/8) with all partners F anti-symplectically 3-congruent to E. These partners form a rational curve (the twist X_E^−(3) of X(3)); its parameter w gives C_w over ℚ(w) with explicit degree-3 maps onto E and F_w, so J is isogenous over ℚ to E × F_w. The map onto E has prime degree, so pullback is injective on E and [8] ⊂ J(ℚ). Every 11- and 13-integral w ≡ 5 (mod 143) has the reductions of w = 5 at 11 and 13, where J(𝔽₁₁) ≅ ℤ/112 and J(𝔽₁₃) ≅ [2,2,40]; the largest group embedding in both is [8], so J(ℚ)_tors is exactly [8]; the G2-invariants at w = 5 and w = −138 differ (family_8_split_certificate.log)", "inherited_from": null } } }, { "key": "9", "group": [ 9 ], "bracket": "[9]", "label": "ℤ/9", "order": 9, "rank": 1, "classes": { "simple": [ { "id": "9.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "0", "0", "0", "-1" ], "h": [ "1", "1", "0", "1" ], "equation": { "f": "-x^4", "h": "x^3 + x + 1", "text": "y^2 + (x^3 + x + 1)*y = -x^4" }, "conductor": "713", "conductor_sort": 713, "conductor_note": "", "discriminant_digits": 7, "lmfdb": { "label": "713.b.713.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/713/b/713/1", "snapshot_label": "" }, "sources": [ "Flynn1991" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": true, "year": null, "historical": false, "first_source": "Flynn1991", "first_source_year": 1991, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 9 (source: \\cite{Flynn1991})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 7 is strict (χ = T^4 + T^3 - 4*T^2 + 7*T + 49)", "verified": "2026-09-18 17:32:15 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "9.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "-3", "1", "-5", "1", "-2", "-1" ], "h": [ "0", "1", "0", "1" ], "equation": { "f": "-x^5 - 2*x^4 + x^3 - 5*x^2 + x - 3", "h": "x^3 + x", "text": "y^2 + (x^3 + x)*y = -x^5 - 2*x^4 + x^3 - 5*x^2 + x - 3" }, "conductor": "722", "conductor_sort": 722, "conductor_note": "", "discriminant_digits": 13, "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" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 9 (source: \\cite{BSSVY})", "new_group": false, "new_for_class": true, "certificate": "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]", "verified": "2026-09-18 17:35:56 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [ { "id": "9.c", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "26", "3", "-12", "9", "2", "-2", "1" ], "h": [ "0", "1" ], "equation": { "f": "x^6 - 2*x^5 + 2*x^4 + 9*x^3 - 12*x^2 + 3*x + 26", "h": "x", "text": "y^2 + x*y = x^6 - 2*x^5 + 2*x^4 + 9*x^3 - 12*x^2 + 3*x + 26" }, "conductor": "1696", "conductor_sort": 1696, "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", "discriminant_digits": 10, "lmfdb": { "label": "1696.b.434176.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/1696/b/434176/1", "snapshot_label": "" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": 2016, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "generators", "reference": "LMFDB genus 2 curve 1696.b.434176.1, https://www.lmfdb.org/Genus2Curve/Q/1696/b/434176/1", "new_group": false, "new_for_class": true, "certificate": "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); simple over ℚ: χ_3 = T^4 - T^2 + 9 is irreducible", "verified": "2026-10-17 06:29:05 UTC", "submitted": "2026-10-16", "source_kind": "file" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": 0, "undecided": null }, "first_source": { "simple": "Flynn1991", "qsplit": "BSSVY", "gsplit": "BSSVY", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite" ], "note": "contact-9 chart h = 1 − (9/2)x + (63/8)x² − (105/16)x³ + tx⁴, f = (h² + (x−1)⁹)/x⁴: [(1, h(1)) − ∞] has order 9 over ℚ(t). Every 3-integral t ≡ 1 (mod 3) has #J(𝔽₃) = 9, strict (family_9_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings, in the normal form of Bröker–Howe–Lauter–Stevenhagen (arXiv:1403.6911), of E = 54b3 (torsion ℤ/9) with all partners F anti-symplectically 3-congruent to E. These partners form a rational curve (the twist X_E^−(3) of X(3)); its parameter w gives C_w over ℚ(w) with explicit degree-3 maps onto E and F_w, so J is isogenous over ℚ to E × F_w. The map onto E has prime degree, so pullback is injective on E and [9] ⊂ J(ℚ); F_w also inherits a rational point of order 3 from E, but no extra 3-torsion appears in J on the class below. Every 5- and 7-integral w ≡ −2 (mod 35) has the reductions of w = −2 at 5 and 7, where J(𝔽₅) ≅ [9,9] and J(𝔽₇) ≅ ℤ/54; the largest group embedding in both is [9], so J(ℚ)_tors is exactly [9]; the G2-invariants at w = −2 and w = 33 differ (family_9_split_certificate.log)", "inherited_from": null } } }, { "key": "10", "group": [ 10 ], "bracket": "[10]", "label": "ℤ/10", "order": 10, "rank": 1, "classes": { "simple": [ { "id": "10.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "7", "16", "0", "-8", "-2", "1" ], "h": [ "0", "1", "0", "1" ], "equation": { "f": "x^5 - 2*x^4 - 8*x^3 + 16*x + 7", "h": "x^3 + x", "text": "y^2 + (x^3 + x)*y = x^5 - 2*x^4 - 8*x^3 + 16*x + 7" }, "conductor": "389", "conductor_sort": 389, "conductor_note": "", "discriminant_digits": 7, "lmfdb": { "label": "389.a.389.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/389/a/389/1", "snapshot_label": "" }, "sources": [ "Flynn1991" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": true, "year": null, "historical": false, "first_source": "Flynn1991", "first_source_year": 1991, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 10 (source: \\cite{Flynn1991})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 5 is strict (χ = T^4 - T^3 - 5*T + 25)", "verified": "2026-09-18 17:32:15 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "10.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "-2", "-1", "10", "7", "-13", "-11" ], "h": [ "0", "0", "1" ], "equation": { "f": "-11*x^5 - 13*x^4 + 7*x^3 + 10*x^2 - x - 2", "h": "x^2", "text": "y^2 + x^2*y = -11*x^5 - 13*x^4 + 7*x^3 + 10*x^2 - x - 2" }, "conductor": "363", "conductor_sort": 363, "conductor_note": "", "discriminant_digits": 5, "lmfdb": { "label": "363.a.43923.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/363/a/43923/1", "snapshot_label": "" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 10 (source: \\cite{BSSVY})", "new_group": false, "new_for_class": true, "certificate": "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]", "verified": "2026-09-18 17:35:57 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [ { "id": "10.c", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "0", "-1", "0", "-1" ], "h": [ "1", "1", "1", "1" ], "equation": { "f": "-x^3 - x", "h": "x^3 + x^2 + x + 1", "text": "y^2 + (x^3 + x^2 + x + 1)*y = -x^3 - x" }, "conductor": "576", "conductor_sort": 576, "conductor_note": "", "discriminant_digits": 7, "lmfdb": { "label": "576.a.576.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/576/a/576/1", "snapshot_label": "" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": 2016, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "LMFDB genus 2 curve 576.a.576.1, https://www.lmfdb.org/Genus2Curve/Q/576/a/576/1", "new_group": false, "new_for_class": true, "certificate": "(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); simple over ℚ: χ_19 = T^4 - 22*T^2 + 361 is irreducible", "verified": "2026-10-17 06:01:48 UTC", "submitted": "2026-10-16", "source_kind": "file" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": 0, "undecided": null }, "first_source": { "simple": "Flynn1991", "qsplit": "BSSVY", "gsplit": "BSSVY", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite" ], "note": "contact-5 chart at b = 1: y² = (1 + tx + x²)² − (t+2)²x⁵, with [x, 1] = [(0,1) − ∞] of order 5 and [x − 1, 0] of order 2 over ℚ(t). Every t ≡ 15 (mod 35) has #J(𝔽₅) = 20 and #J(𝔽₇) = 30 (gcd 10), strict at 5 and 7 (family_10_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings, in the normal form of Bröker–Howe–Lauter–Stevenhagen (arXiv:1403.6911), of E = 66c1 (torsion ℤ/10) with all partners F anti-symplectically 3-congruent to E. These partners form a rational curve (the twist X_E^−(3) of X(3)); its parameter w gives C_w over ℚ(w) with explicit degree-3 maps onto E and F_w, so J is isogenous over ℚ to E × F_w. The map onto E has prime degree, so pullback is injective on E and [10] ⊂ J(ℚ). Every 7- and 13-integral w ≡ 1 (mod 91) has the reductions of w = 1 at 7 and 13, where J(𝔽₇) ≅ [2,20] and J(𝔽₁₃) ≅ ℤ/190; the largest group embedding in both is [10], so J(ℚ)_tors is exactly [10]; the G2-invariants at w = 1 and w = −90 differ (family_10_split_certificate.log)", "inherited_from": null } } }, { "key": "11", "group": [ 11 ], "bracket": "[11]", "label": "ℤ/11", "order": 11, "rank": 1, "classes": { "simple": [ { "id": "11.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "0", "1" ], "h": [ "1", "1", "0", "1" ], "equation": { "f": "x^2", "h": "x^3 + x + 1", "text": "y^2 + (x^3 + x + 1)*y = x^2" }, "conductor": "353", "conductor_sort": 353, "conductor_note": "", "discriminant_digits": 7, "lmfdb": { "label": "353.a.353.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/353/a/353/1", "snapshot_label": "" }, "sources": [ "Ogg1973" ], "rm": false, "discovered_by": "Platonov–Petrunin 2012a", "credit_differs": true, "year": null, "historical": false, "first_source": "Ogg1973", "first_source_year": 1973, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 11 (source: \\cite{Ogg1973})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 3 is strict (χ = T^4 + 2*T^3 + 4*T^2 + 6*T + 9)", "verified": "2026-09-18 17:32:16 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [], "gsplit": [ { "id": "11.b", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "9471400", "94188680", "99564424", "-65405928", "31412066", "-5109634", "1204142" ], "h": [ "0" ], "equation": { "f": "1204142*x^6 - 5109634*x^5 + 31412066*x^4 - 65405928*x^3 + 99564424*x^2 + 94188680*x + 9471400", "h": "0", "text": "y^2 = 1204142*x^6 - 5109634*x^5 + 31412066*x^4 - 65405928*x^3 + 99564424*x^2 + 94188680*x + 9471400" }, "conductor": "", "conductor_sort": null, "conductor_note": "not computed: Magma's genus-2 Conductor failed on this curve: Fibre blowups over number fields not yet implemented", "discriminant_digits": 83, "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": "" }, "sources": [ "BNSS2026" ], "rm": false, "discovered_by": "Balakrishnan–Najman–Shnidman–Sutherland 2026", "credit_differs": false, "year": null, "historical": false, "first_source": "BNSS2026", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 11 (source: new)", "new_group": false, "new_for_class": true, "certificate": "(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); simple over ℚ: χ_3 = T^4 + T^2 + 9 is irreducible", "verified": "2026-09-18 17:35:59 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "Ogg1973", "qsplit": null, "gsplit": "BNSS2026", "undecided": null }, "known": { "simple": true, "qsplit": false, "gsplit": true, "undecided": false }, "exclusions": { "qsplit": { "reason": "A Jacobian split over ℚ has no rational torsion of prime order greater than 7, by Mazur's theorem applied to an elliptic subvariety and its elliptic quotient.", "sources": [ "QSplitPrimeObstruction", "Mazur1977" ], "proof_url": "proofs/qsplit-prime-obstruction.html" } }, "n_curves": 2, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite", "DaowsudSchmidt" ], "note": "Flynn's family y² = x⁶ + 2x⁵ + (2t+3)x⁴ + 2x³ + (t²+1)x² + 2t(1−t)x + t² (as quoted by Daowsud–Schmidt): [∞₋ − ∞₊] has order 11 over ℚ(t). Every t ≡ 11 (mod 15) has #J(𝔽₃) = 11 and #J(𝔽₅) = 22, strict at 5 (family_11_certificate.log). The record also cites the Daowsud–Schmidt continued-fraction family, not certified here", "inherited_from": null }, "split": { "grade": "open", "sources": [ "BNSS2026", "ThisSite" ], "note": "a single curve is known, 11.b (BNSS2026): the (2,2)-gluing of an elliptic curve E over ℚ(√11) with a point of order 11 to its Galois conjugate E^σ. The gluing needs E[2] ≅ E^σ[2] over the quadratic field, which is not automatic. The curves E over quadratic fields with a point of order 11 are parametrised by s ∈ ℚ (via X₁(11)), and the s for which E[2] ≅ E^σ[2] are the images of the rational points of a curve Y of genus at least 8 (a degree-6 cover of the s-line whose reductions modulo 101, 103 and 107 are absolutely irreducible of genus 8). So this construction gives only finitely many curves; to height 400 in s the only one is 11.b (split_11_remark.log; a remark, not a certificate)", "inherited_from": null } } }, { "key": "12", "group": [ 12 ], "bracket": "[12]", "label": "ℤ/12", "order": 12, "rank": 1, "classes": { "simple": [ { "id": "12.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "1", "1", "1" ], "h": [ "0", "1", "1", "1" ], "equation": { "f": "x^2 + x + 1", "h": "x^3 + x^2 + x", "text": "y^2 + (x^3 + x^2 + x)*y = x^2 + x + 1" }, "conductor": "762", "conductor_sort": 762, "conductor_note": "", "discriminant_digits": 8, "lmfdb": { "label": "762.a.3048.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/762/a/3048/1", "snapshot_label": "" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 12 (source: \\cite{BSSVY})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 7 is strict (χ = T^4 - T^3 + 6*T^2 - 7*T + 49)", "verified": "2026-09-18 17:32:16 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "12.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "0", "0", "1", "-1", "1" ], "h": [ "1", "0", "0", "1" ], "equation": { "f": "x^4 - x^3 + x^2", "h": "x^3 + 1", "text": "y^2 + (x^3 + 1)*y = x^4 - x^3 + x^2" }, "conductor": "294", "conductor_sort": 294, "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)", "discriminant_digits": 7, "lmfdb": { "label": "294.a.294.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/294/a/294/1", "snapshot_label": "" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 12 (source: \\cite{BSSVY})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:36:00 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [ { "id": "12.c", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "0", "-2", "1", "0", "-1" ], "h": [ "1", "1", "1", "1" ], "equation": { "f": "-x^4 + x^2 - 2*x", "h": "x^3 + x^2 + x + 1", "text": "y^2 + (x^3 + x^2 + x + 1)*y = -x^4 + x^2 - 2*x" }, "conductor": "1792", "conductor_sort": 1792, "conductor_note": "", "discriminant_digits": 8, "lmfdb": { "label": "1792.a.3584.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/1792/a/3584/1", "snapshot_label": "" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": 2016, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "generators", "reference": "LMFDB genus 2 curve 1792.a.3584.1, https://www.lmfdb.org/Genus2Curve/Q/1792/a/3584/1", "new_group": false, "new_for_class": true, "certificate": "(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); simple over ℚ: χ_5 = T^4 - 2*T^2 + 25 is irreducible", "verified": "2026-10-17 06:01:49 UTC", "submitted": "2026-10-16", "source_kind": "file" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": 0, "undecided": null }, "first_source": { "simple": "BSSVY", "qsplit": "BSSVY", "gsplit": "BSSVY", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite" ], "note": "the rational-Weierstrass subchart a = (1−z²)/(4(r+1)) of the M(12) chart, slice r = 3: T₁₂ + 1 has the rational root w = 8/(1−z), and [P − (w, 0)] has order 12 over ℚ(z). Every z ≡ 37 (mod 77) has #J(𝔽₇) = 60 and #J(𝔽₁₁) = 168 (gcd 12), strict at 11 (family_12_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings, in the normal form of Bröker–Howe–Lauter–Stevenhagen (arXiv:1403.6911), of E = 90c3 (torsion ℤ/12) with all partners F anti-symplectically 3-congruent to E. These partners form a rational curve (the twist X_E^−(3) of X(3)); its parameter w gives C_w over ℚ(w) with explicit degree-3 maps onto E and F_w, so J is isogenous over ℚ to E × F_w. The map onto E has prime degree, so pullback is injective on E and [12] ⊂ J(ℚ); F_w also inherits a rational point of order 3 from E, but no extra 3-torsion appears in J on the class below. Every 11- and 13-integral w ≡ −16 (mod 143) has the reductions of w = −16 at 11 and 13, where J(𝔽₁₁) ≅ ℤ/180 and J(𝔽₁₃) ≅ [3,84]; the largest group embedding in both is [12], so J(ℚ)_tors is exactly [12]; the G2-invariants at w = −16 and w = 127 differ (family_12_split_certificate.log)", "inherited_from": null } } }, { "key": "13", "group": [ 13 ], "bracket": "[13]", "label": "ℤ/13", "order": 13, "rank": 1, "classes": { "simple": [ { "id": "13.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "0", "-1", "-1" ], "h": [ "1", "1", "1", "1" ], "equation": { "f": "-x^3 - x^2", "h": "x^3 + x^2 + x + 1", "text": "y^2 + (x^3 + x^2 + x + 1)*y = -x^3 - x^2" }, "conductor": "349", "conductor_sort": 349, "conductor_note": "", "discriminant_digits": 7, "lmfdb": { "label": "349.a.349.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/349/a/349/1", "snapshot_label": "" }, "sources": [ "Flynn1990", "Leprevost1991a" ], "rm": false, "discovered_by": "Flynn 1990; Leprévost 1991 (order 13)", "credit_differs": false, "year": null, "historical": false, "first_source": "Flynn1990", "first_source_year": 1990, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 13 (source: \\cite{Flynn1990,Leprevost1991a})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 3 is strict (χ = T^4 + T^3 - T^2 + 3*T + 9)", "verified": "2026-09-18 17:32:17 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "Flynn1990", "qsplit": null, "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": false, "gsplit": false, "undecided": false }, "exclusions": { "qsplit": { "reason": "A Jacobian split over ℚ has no rational torsion of prime order greater than 7, by Mazur's theorem applied to an elliptic subvariety and its elliptic quotient.", "sources": [ "QSplitPrimeObstruction", "Mazur1977" ], "proof_url": "proofs/qsplit-prime-obstruction.html" }, "gsplit": { "reason": "A geometrically split genus-2 Jacobian over ℚ with a rational point of order 13 is not simple over ℚ. The proof applies to every geometric splitting degree and uses the principal polarization.", "sources": [ "QSimple13Obstruction", "BBDN2014", "Vukorepa2022", "Box2021", "Mazur1977", "Kenku1979", "Kenku1980", "Faltings1983", "SerreTate1968", "Tate1966" ], "proof_url": "proofs/qsimple-13-obstruction.html" } }, "n_curves": 1, "infinite": { "simple": { "grade": "exact", "sources": [ "ThisSite" ], "note": "relation-lattice family: B − A = x(x−1)², B + A = (x−1)³ + tx², f_t = A² + tx³(x−1)² = B² − x(x−1)⁵; [P₁ − ∞] has order 13 over ℚ(t) (lattice (3,2),(1,5)). Every 3-integral t ≡ 1 (mod 3) has the reduction of t = 1 at 3: #J(𝔽₃) = 13, strict (family_13_certificate.log)", "inherited_from": null }, "split": { "grade": "impossible", "sources": [ "QSimple13Obstruction", "BBDN2014", "Vukorepa2022", "Box2021", "Mazur1977", "Kenku1979", "Kenku1980", "Faltings1983", "SerreTate1968", "Tate1966", "QSplitPrimeObstruction" ], "note": "A Jacobian split over ℚ has no rational torsion of prime order greater than 7, by Mazur's theorem applied to an elliptic subvariety and its elliptic quotient. A geometrically split genus-2 Jacobian over ℚ with a rational point of order 13 is not simple over ℚ. The proof applies to every geometric splitting degree and uses the principal polarization.", "proof_url": "proofs/qsimple-13-obstruction.html", "inherited_from": null } } }, { "key": "14", "group": [ 14 ], "bracket": "[14]", "label": "ℤ/14", "order": 14, "rank": 1, "classes": { "simple": [ { "id": "14.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "1", "1" ], "h": [ "1", "0", "0", "1" ], "equation": { "f": "x^2 + x", "h": "x^3 + 1", "text": "y^2 + (x^3 + 1)*y = x^2 + x" }, "conductor": "249", "conductor_sort": 249, "conductor_note": "", "discriminant_digits": 7, "lmfdb": { "label": "249.a.249.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/249/a/249/1", "snapshot_label": "" }, "sources": [ "PP2012", "PZP2013" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": true, "year": null, "historical": false, "first_source": "PP2012", "first_source_year": 2012, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 14 (source: \\cite{PP2012,PZP2013})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 7 is strict (χ = T^4 + T^3 - 2*T^2 + 7*T + 49)", "verified": "2026-09-18 17:32:17 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "14.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "0", "-4", "1", "-3", "1", "-1" ], "h": [ "0", "0", "1", "1" ], "equation": { "f": "-x^5 + x^4 - 3*x^3 + x^2 - 4*x", "h": "x^3 + x^2", "text": "y^2 + (x^3 + x^2)*y = -x^5 + x^4 - 3*x^3 + x^2 - 4*x" }, "conductor": "1352", "conductor_sort": 1352, "conductor_note": "", "discriminant_digits": 10, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 13 (source: \\cite{BookerSutherland})", "new_group": false, "new_for_class": true, "certificate": "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]", "verified": "2026-09-18 17:36:01 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [ { "id": "14.c", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "15", "9", "-7", "64", "47", "-27", "3" ], "h": [ "0", "1", "1" ], "equation": { "f": "3*x^6 - 27*x^5 + 47*x^4 + 64*x^3 - 7*x^2 + 9*x + 15", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = 3*x^6 - 27*x^5 + 47*x^4 + 64*x^3 - 7*x^2 + 9*x + 15" }, "conductor": "9604", "conductor_sort": 9604, "conductor_note": "", "discriminant_digits": 20, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": 2026, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "generators", "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)", "new_group": false, "new_for_class": true, "certificate": "a chain of at most 2 (2,2)-isogenies over Q(sqrt(-7)) reaches a product of two elliptic curves over Q(sqrt(-7)); simple over ℚ: χ_5 = T^4 + 2*T^2 + 25 is irreducible", "verified": "2026-10-17 06:01:50 UTC", "submitted": "2026-10-16", "source_kind": "file" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": 0, "undecided": null }, "first_source": { "simple": "PP2012", "qsplit": "BookerSutherland", "gsplit": "BookerSutherland", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite" ], "note": "rational-root subfamily of the contact-7 chart: r = 1 − s², h(r) = s⁷, slice b = 1. [(1, h(1)) − ∞] has order 7 and [(r, 0) − ∞] order 2 over ℚ(s). Every s ≡ 31 (mod 35) has #J(𝔽₅) = 28 and #J(𝔽₇) = 70 (gcd 14), strict at 7 (family_14_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings, in the normal form of Bröker–Howe–Lauter–Stevenhagen (arXiv:1403.6911), of E = 26b1 (torsion ℤ/7) with the partners F that have a rational 2-torsion point. The partners anti-symplectically 3-congruent to E form a rational curve, and so does the cover on which F has a rational 2-torsion point (Γ(3) ∩ Γ₀(2) has genus 0); this gives C_v over ℚ(v) with explicit degree-3 maps onto E and F_v, so J is isogenous over ℚ to E × F_v. Pullback is injective on torsion of order prime to 3 (φ_*φ^* = [3]), so ℤ/14 ⊂ J(ℚ). Every 7- and 71-integral v ≡ 0 (mod 497) has the reductions of v = 0 (the census curve 14.b, the gluing of 26b1 and 52a1) at 7 and 71, where #J(𝔽₇) = 70 and #J(𝔽₇₁) = 4774, so J(ℚ)_tors is exactly [14]; the G2-invariants at v = 0 and v = 497 differ (family_14_split_certificate.log)", "inherited_from": null } } }, { "key": "15", "group": [ 15 ], "bracket": "[15]", "label": "ℤ/15", "order": 15, "rank": 1, "classes": { "simple": [ { "id": "15.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "-1", "-1" ], "h": [ "1", "1", "1", "1" ], "equation": { "f": "-x^2 - x", "h": "x^3 + x^2 + x + 1", "text": "y^2 + (x^3 + x^2 + x + 1)*y = -x^2 - x" }, "conductor": "277", "conductor_sort": 277, "conductor_note": "", "discriminant_digits": 7, "lmfdb": { "label": "277.a.277.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/277/a/277/1", "snapshot_label": "" }, "sources": [ "Leprevost1991" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": true, "year": null, "historical": false, "first_source": "Leprevost1991", "first_source_year": 1991, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 15 (source: \\cite{Leprevost1991})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 3 is strict (χ = T^4 + T^3 + T^2 + 3*T + 9)", "verified": "2026-09-18 17:32:18 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "15.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "1", "0", "2", "0", "1" ], "h": [ "0", "0", "0", "1" ], "equation": { "f": "x^4 + 2*x^2 + 1", "h": "x^3", "text": "y^2 + x^3*y = x^4 + 2*x^2 + 1" }, "conductor": "484", "conductor_sort": 484, "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)", "discriminant_digits": 7, "lmfdb": { "label": "484.a.1936.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/484/a/1936/1", "snapshot_label": "" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 14 (source: \\cite{BSSVY})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:36:03 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [ { "id": "15.c", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "1", "-18", "72", "47", "38", "9", "1" ], "h": [ "0", "0", "1" ], "equation": { "f": "x^6 + 9*x^5 + 38*x^4 + 47*x^3 + 72*x^2 - 18*x + 1", "h": "x^2", "text": "y^2 + x^2*y = x^6 + 9*x^5 + 38*x^4 + 47*x^3 + 72*x^2 - 18*x + 1" }, "conductor": "3168", "conductor_sort": 3168, "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", "discriminant_digits": 20, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": 2026, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "generators", "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)", "new_group": false, "new_for_class": true, "certificate": "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); simple over ℚ: χ_5 = T^4 + 4*T^2 + 25 is irreducible", "verified": "2026-10-17 06:01:53 UTC", "submitted": "2026-10-16", "source_kind": "file" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": 0, "undecided": null }, "first_source": { "simple": "Leprevost1991", "qsplit": "BSSVY", "gsplit": "BookerSutherland", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "ThisSite" ], "note": "relation-lattice family: B − A = tx² + (x−1)³, B + A = x³, f_t = A² + tx⁵ = B² − x³(x−1)³; a class of order 15 over ℚ(t) (lattice (5,0),(−3,3)). Every 3-integral t ≡ 1 (mod 3) has #J(𝔽₃) = 15, strict (family_15_certificate.log). (The record's [15] was only inherited from [30], i.e. containment)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite", "HLP2000" ], "note": "the construction of the ℤ/21 entry of HLP 2000 (Lemma 11 and Prop. 4) with the Kubert curve E₅(t) (torsion ℤ/5; short model y² = x³ + Ax + B, A = −27c₄, B = −54c₆) in place of E₇: E₃: y² = x³ + (x − B²/A³)² has a point of order 3 and 2-torsion Galois-isomorphic to that of E₅, and C_t: y² = A⁷ − 3A⁵Bx² + (A⁶ + 3A³B²)x⁴ − AB³x⁶ maps onto E₃ by (x, y) ↦ (x², y) and onto E₅ by (x, y) ↦ (1/x², y/x³), so J is isogenous over ℚ to E₃ × E₅. The pullbacks of the points of order 5 and 3 mark [15] over ℚ(t). Every 7- and 11-integral t ≡ 4 (mod 77) has #J(𝔽₇) = 60 and #J(𝔽₁₁) = 225, so J(ℚ)_tors is exactly [15]; the G2-invariants of the members differ (family_15_split_certificate.log)", "inherited_from": null } } }, { "key": "16", "group": [ 16 ], "bracket": "[16]", "label": "ℤ/16", "order": 16, "rank": 1, "classes": { "simple": [ { "id": "16.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "1", "1", "-2", "0", "1", "-1" ], "h": [ "1", "0", "0", "1" ], "equation": { "f": "-x^5 + x^4 - 2*x^2 + x + 1", "h": "x^3 + 1", "text": "y^2 + (x^3 + 1)*y = -x^5 + x^4 - 2*x^2 + x + 1" }, "conductor": "830", "conductor_sort": 830, "conductor_note": "", "discriminant_digits": 8, "lmfdb": { "label": "830.a.6640.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/830/a/6640/1", "snapshot_label": "" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 16 (source: \\cite{BSSVY})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 3 is strict (χ = T^4 + T^3 + 2*T^2 + 3*T + 9)", "verified": "2026-09-18 17:32:18 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "16.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "-125", "0", "97", "0", "-26", "0", "2" ], "h": [ "0", "1", "0", "1" ], "equation": { "f": "2*x^6 - 26*x^4 + 97*x^2 - 125", "h": "x^3 + x", "text": "y^2 + (x^3 + x)*y = 2*x^6 - 26*x^4 + 97*x^2 - 125" }, "conductor": "2700", "conductor_sort": 2700, "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)", "discriminant_digits": 13, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 15 (source: \\cite{BookerSutherland})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:36:04 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "BSSVY", "qsplit": "BookerSutherland", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 2, "infinite": { "simple": { "grade": "exact", "sources": [ "ThisSite" ], "note": "relation-lattice family: λ = −(t−1)²/t, B − A = (x−1)²(x−t), B + A = x²·(λx + (x−1)²)/(x−t), f_t = A² + λx³(x−1)² = B² − x²(x−1)⁴; a cyclic class of order 16 over ℚ(t) (lattice (3,2),(−2,4)). Every 3-integral t ≡ 2 (mod 3) has #J(𝔽₃) = 16, strict (family_16_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4) of E₁ = 15a4 (torsion ℤ/8, P = (2, 6); 2-torsion field ℚ(i), roots 3/4 and θ₁ = −1 ± 6i in the monic model) with the curves E₂(s): y² = x(x² + (6 + 24s − 6s²)x + 45(s² + 1)²) = x(x − θ₂)(x − θ₂′), θ₂ = −(3 − 6i)(1 + si)², along (3/4, θ₁, θ₁′) ↦ (0, θ₂, θ₂′): the Kummer class of (0, 0) on E₂(s) equals that of P (−θ₂ = (2 − θ₁)(1 + si)²), which is the gain of the gluing. Up to a square C_s: y² = (s² − 11s − 1)((s² + s − 1)x² + 1)(144(s² + 1)²x⁴ + 24(11s² + 4s − 11)x² + 125), with (x, y) ↦ (x², y) onto E₂(s) and (x, y) ↦ (1/x², y/x³) onto E₁, so J is isogenous over ℚ to E₁ × E₂(s). The sum of the pullbacks of a half of P and a half of (0, 0), computed over a field of degree 8, has coefficients in ℚ(s) and order 16, so ℤ/16 is marked over ℚ(s). Every 7- and 23-integral s ≡ 1 (mod 161) has #J(𝔽₇) = 80 and #J(𝔽₂₃) = 432, so J(ℚ)_tors is exactly [16]; the G2-invariants of the members differ. The member s = 0 (E₂ = 180a2) is the census curve 16.b (family_16_split_certificate.log)", "inherited_from": null } } }, { "key": "17", "group": [ 17 ], "bracket": "[17]", "label": "ℤ/17", "order": 17, "rank": 1, "classes": { "simple": [ { "id": "17.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "1", "3", "1", "1" ], "h": [ "0", "1", "1", "1" ], "equation": { "f": "x^3 + x^2 + 3*x + 1", "h": "x^3 + x^2 + x", "text": "y^2 + (x^3 + x^2 + x)*y = x^3 + x^2 + 3*x + 1" }, "conductor": "1996", "conductor_sort": 1996, "conductor_note": "", "discriminant_digits": 10, "lmfdb": { "label": "1996.b.510976.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/1996/b/510976/1", "snapshot_label": "" }, "sources": [ "Leprevost1991" ], "rm": false, "discovered_by": "Leprévost 1991 (orders 15, 17, 19, 21)", "credit_differs": false, "year": null, "historical": false, "first_source": "Leprevost1991", "first_source_year": 1991, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 17 (source: \\cite{Leprevost1991})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 3 is strict (χ = T^4 + T^3 + 3*T^2 + 3*T + 9)", "verified": "2026-09-18 17:32:19 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "Leprevost1991", "qsplit": null, "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": false, "gsplit": false, "undecided": false }, "exclusions": { "qsplit": { "reason": "A Jacobian split over ℚ has no rational torsion of prime order greater than 7, by Mazur's theorem applied to an elliptic subvariety and its elliptic quotient.", "sources": [ "QSplitPrimeObstruction", "Mazur1977" ], "proof_url": "proofs/qsplit-prime-obstruction.html" }, "gsplit": { "reason": "A geometrically split abelian surface over ℚ has no rational point of order 17. The proof applies to every geometric splitting degree.", "sources": [ "Geometric17Obstruction", "Mazur1977", "Kamienny1992", "SerreTate1968" ], "proof_url": "proofs/geometric-17-obstruction.html" } }, "n_curves": 1, "infinite": { "simple": { "grade": "exact", "sources": [ "ThisSite" ], "note": "relation-lattice family: λ = −(t−1)³/t², B − A = (x−1)²(x−t), B + A = x·((x−1)³ + λx²)/(x−t), f_t = A² + λx³(x−1)² = B² − x(x−1)⁵; a class of order 17 over ℚ(t) (lattice (3,2),(−1,5)). Every 3-integral t ≡ 2 (mod 3) has #J(𝔽₃) = 17, strict (family_17_certificate.log)", "inherited_from": null }, "split": { "grade": "impossible", "sources": [ "Geometric17Obstruction", "Mazur1977", "Kamienny1992", "SerreTate1968" ], "note": "A geometrically split abelian surface over ℚ has no rational point of order 17. The proof applies to every geometric splitting degree.", "proof_url": "proofs/geometric-17-obstruction.html", "inherited_from": null } } }, { "key": "18", "group": [ 18 ], "bracket": "[18]", "label": "ℤ/18", "order": 18, "rank": 1, "classes": { "simple": [ { "id": "18.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "2", "4", "0", "-2" ], "h": [ "1", "0", "0", "1" ], "equation": { "f": "-2*x^4 + 4*x^2 + 2*x", "h": "x^3 + 1", "text": "y^2 + (x^3 + 1)*y = -2*x^4 + 4*x^2 + 2*x" }, "conductor": "1180", "conductor_sort": 1180, "conductor_note": "", "discriminant_digits": 8, "lmfdb": { "label": "1180.a.18880.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/1180/a/18880/1", "snapshot_label": "" }, "sources": [ "PP2012", "PZP2013" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": true, "year": null, "historical": false, "first_source": "PP2012", "first_source_year": 2012, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 18 (source: \\cite{PP2012,PZP2013})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 7 is strict (χ = T^4 + T^3 - 4*T^2 + 7*T + 49)", "verified": "2026-09-18 17:32:19 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "18.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "-6", "5", "-7", "2", "-1", "-1" ], "h": [ "0", "0", "1", "1" ], "equation": { "f": "-x^5 - x^4 + 2*x^3 - 7*x^2 + 5*x - 6", "h": "x^3 + x^2", "text": "y^2 + (x^3 + x^2)*y = -x^5 - x^4 + 2*x^3 - 7*x^2 + 5*x - 6" }, "conductor": "490", "conductor_sort": 490, "conductor_note": "", "discriminant_digits": 10, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 16 (source: \\cite{BookerSutherland})", "new_group": false, "new_for_class": true, "certificate": "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]", "verified": "2026-09-18 17:36:06 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [ { "id": "18.c", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "10", "-16", "-14", "5", "0", "-1" ], "h": [ "0", "1", "0", "1" ], "equation": { "f": "-x^5 + 5*x^3 - 14*x^2 - 16*x + 10", "h": "x^3 + x", "text": "y^2 + (x^3 + x)*y = -x^5 + 5*x^3 - 14*x^2 - 16*x + 10" }, "conductor": "17280", "conductor_sort": 17280, "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", "discriminant_digits": 15, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": 2026, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "generators", "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)", "new_group": false, "new_for_class": true, "certificate": "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); simple over ℚ: χ_7 = T^4 + 4*T^2 + 49 is irreducible", "verified": "2026-10-17 10:20:31 UTC", "submitted": "2026-10-17", "source_kind": "file" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": 0, "undecided": null }, "first_source": { "simple": "PP2012", "qsplit": "BookerSutherland", "gsplit": "BookerSutherland", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite" ], "note": "rational-root subfamily of the contact-9 chart: r = 1 − s², h(r) = s⁹. [(1, h(1)) − ∞] has order 9 and [(r, 0) − ∞] order 2 over ℚ(s). Every s ≡ 23 (mod 91) has #J(𝔽₇) = 72 and #J(𝔽₁₃) = 198 (gcd 18), strict at 13 (family_18_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings, in the normal form of Bröker–Howe–Lauter–Stevenhagen (arXiv:1403.6911), of E = 54b3 (torsion ℤ/9) with the partners F that have a rational 2-torsion point. The partners anti-symplectically 3-congruent to E form a rational curve, and so does the cover on which F has a rational 2-torsion point (Γ(3) ∩ Γ₀(2) has genus 0); this gives C_v over ℚ(v) with explicit degree-3 maps onto E and F_v, so J is isogenous over ℚ to E × F_v. The map onto E has prime degree, so pullback is injective on E and ℤ/9 ⊂ J(ℚ); with the class of the quadratic factor of f_v, [18] is marked. F_v also inherits a rational point of order 3 from E, but no extra 3-torsion appears in J (ℤ/18 is not a torsion group of an elliptic curve over ℚ). Here #J(𝔽_p) = #E(𝔽_p)#F_v(𝔽_p) is always divisible by 54, so the group structure is used instead: every 5- and 13-integral v ≡ −2 (mod 65) has the reductions of v = −2 at 5 and 13, where J(𝔽₅) ≅ ℤ/54 is cyclic and J(𝔽₁₃) ≅ (ℤ/18)² has exponent 18, so J(ℚ)_tors is exactly [18]; the G2-invariants at v = −2 and v = 63 differ. The member v = 4 (outside the class) is the census curve 36.b (family_18_split_certificate.log)", "inherited_from": null } } }, { "key": "19", "group": [ 19 ], "bracket": "[19]", "label": "ℤ/19", "order": 19, "rank": 1, "classes": { "simple": [ { "id": "19.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "-2", "5", "-3", "-5", "1", "1", "1" ], "h": [ "1", "1", "1" ], "equation": { "f": "x^6 + x^5 + x^4 - 5*x^3 - 3*x^2 + 5*x - 2", "h": "x^2 + x + 1", "text": "y^2 + (x^2 + x + 1)*y = x^6 + x^5 + x^4 - 5*x^3 - 3*x^2 + 5*x - 2" }, "conductor": "1468", "conductor_sort": 1468, "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", "discriminant_digits": 11, "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" }, "sources": [ "Leprevost1991", "PZP2013" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "Leprevost1991", "first_source_year": 1991, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 19 (source: \\cite{Leprevost1991,PZP2013})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 3 is strict (χ = T^4 + T^3 + 5*T^2 + 3*T + 9)", "verified": "2026-09-18 17:32:20 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [], "gsplit": [ { "id": "19.b", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "0", "1", "1" ], "h": [ "1", "0", "1", "1" ], "equation": { "f": "x^2 + x", "h": "x^3 + x^2 + 1", "text": "y^2 + (x^3 + x^2 + 1)*y = x^2 + x" }, "conductor": "169", "conductor_sort": 169, "conductor_note": "", "discriminant_digits": 6, "lmfdb": { "label": "169.a.169.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/169/a/169/1", "snapshot_label": "" }, "sources": [ "MazurTate1973", "Ogg1973" ], "rm": false, "discovered_by": "Mazur–Tate 1973; Ogg 1973", "credit_differs": false, "year": null, "historical": false, "first_source": "MazurTate1973", "first_source_year": 1973, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 17 (source: \\cite{MazurTate1973,Ogg1973})", "new_group": false, "new_for_class": true, "certificate": "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]; simple over ℚ: χ_3 = T^4 + 2*T^3 + T^2 + 6*T + 9 is irreducible", "verified": "2026-09-18 17:36:07 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "Leprevost1991", "qsplit": null, "gsplit": "MazurTate1973", "undecided": null }, "known": { "simple": true, "qsplit": false, "gsplit": true, "undecided": false }, "exclusions": { "qsplit": { "reason": "A Jacobian split over ℚ has no rational torsion of prime order greater than 7, by Mazur's theorem applied to an elliptic subvariety and its elliptic quotient.", "sources": [ "QSplitPrimeObstruction", "Mazur1977" ], "proof_url": "proofs/qsplit-prime-obstruction.html" } }, "n_curves": 2, "infinite": { "simple": { "grade": "exact", "sources": [ "ThisSite", "Leprevost1991" ], "note": "the rational one-parameter family F1: y² = f_r, λ = −r⁴/(r−1)³, S = (λ(x−1)³ + x⁴)/(x−r), A = (S − x(x−1)(x−r))/2, f_r = A² + λx(x−1)⁴ = B² − x⁵(x−1) with B = A + x(x−1)(x−r); P₀ = (0, A(0)), P₁ = (1, A(1)) satisfy P₀ + 4P₁ ~ 5∞ and 5P₀ + P₁ ~ 6∞ (determinant −19), so [P₀ − ∞] has order 19 over ℚ(r). Census curve 19.a is the fiber r = −1. For every 3-integral r ≡ 2 (mod 3) the curve has the same good reduction at 3 as r = 2, with #J(𝔽₃) = 19 and a strict Frobenius, so its torsion is exactly [19] and its Jacobian is geometrically simple; the G2-invariants at r = 2 and r = 5 differ (family_19_certificate.log). Leprévost (1991) gave a family with a class of order 19; whether it is this one was not checked", "inherited_from": null }, "split": { "grade": "open", "sources": [], "note": "", "inherited_from": null } } }, { "key": "20", "group": [ 20 ], "bracket": "[20]", "label": "ℤ/20", "order": 20, "rank": 1, "classes": { "simple": [ { "id": "20.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "0", "0", "0", "0", "-1" ], "h": [ "1", "1" ], "equation": { "f": "-x^5", "h": "x + 1", "text": "y^2 + (x + 1)*y = -x^5" }, "conductor": "394", "conductor_sort": 394, "conductor_note": "", "discriminant_digits": 6, "lmfdb": { "label": "394.a.3152.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/394/a/3152/1", "snapshot_label": "" }, "sources": [ "Leprevost1997" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": true, "year": null, "historical": false, "first_source": "Leprevost1997", "first_source_year": 1997, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 20 (source: \\cite{Leprevost1997})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 7 is strict (χ = T^4 - T^3 - 2*T^2 - 7*T + 49)", "verified": "2026-09-18 17:32:21 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "20.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "-2", "-7", "0", "3", "-2", "1" ], "h": [ "0", "1", "0", "1" ], "equation": { "f": "x^5 - 2*x^4 + 3*x^3 - 7*x - 2", "h": "x^3 + x", "text": "y^2 + (x^3 + x)*y = x^5 - 2*x^4 + 3*x^3 - 7*x - 2" }, "conductor": "363", "conductor_sort": 363, "conductor_note": "", "discriminant_digits": 14, "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" }, "sources": [ "HLP2000" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "HLP2000", "first_source_year": 2000, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 18 (source: \\cite{HLP2000})", "new_group": false, "new_for_class": true, "certificate": "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]", "verified": "2026-09-18 17:36:08 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "Leprevost1997", "qsplit": "HLP2000", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 2, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite" ], "note": "the halving component b = (a²−1)/2 of the contact-5 chart: h = 1 + tx + ((t²−1)/2)x², f_t = h² − ((t+1)⁴/4)x⁵. Over ℚ(t), D₅ = [x, 1] has order 5 and H = [x² + 2x/(t+1), (t+2)x + 1] has order 4 (2H = [x − 1, 0]), so ℤ/20 is marked. Every t ≡ 1 (mod 21) has #J(𝔽₃) = 20 and #J(𝔽₇) = 40 (gcd 20), strict at 7 (family_20_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite", "HLP2000" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4) of E₁ = 66c1 (torsion ℤ/10, non-rational 2-torsion over ℚ(√33)) with the Kubert curve E₂ = E₄(u), u = (33s² − 1)/16, so that 16u + 1 = 33s² and the 2-torsion fields agree (the ℤ/20 entry of HLP 2000, Table 1, on a line of its ℙ²), along the Galois isomorphism of 2-torsion fixing the rational points. The HLP sextic has coefficients in ℚ(s); up to scaling it is C_s: y² = F(x²), with (x, y) ↦ (x², y) onto E₂ and (x, y) ↦ (1/x², y/x³) onto E₁, so J is isogenous over ℚ to E₁ × E₂. Over ℚ(s) the pullbacks of the points of order 10 and 4 (glued along their 2-torsion) mark [20]. Here #J(𝔽_p) is always divisible by 40, so the group structure is used: every 5- and 19-integral s ≡ 2 (mod 95) has J(𝔽₅) ≅ ℤ/80 (cyclic) and J(𝔽₁₉) ≅ [20,20] (exponent 20), so J(ℚ)_tors is exactly [20]; the G2-invariants of the members differ. The member s = 1 (outside the class) has torsion [40], a 2-power gain of the gluing (family_20_split_certificate.log)", "inherited_from": null } } }, { "key": "21", "group": [ 21 ], "bracket": "[21]", "label": "ℤ/21", "order": 21, "rank": 1, "classes": { "simple": [ { "id": "21.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "1", "2", "0", "-1" ], "h": [ "1", "1", "0", "1" ], "equation": { "f": "-x^4 + 2*x^2 + x", "h": "x^3 + x + 1", "text": "y^2 + (x^3 + x + 1)*y = -x^4 + 2*x^2 + x" }, "conductor": "388", "conductor_sort": 388, "conductor_note": "", "discriminant_digits": 7, "lmfdb": { "label": "388.a.776.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/388/a/776/1", "snapshot_label": "" }, "sources": [ "Leprevost1991" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": true, "year": null, "historical": false, "first_source": "Leprevost1991", "first_source_year": 1991, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 21 (source: \\cite{Leprevost1991})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 5 is strict (χ = T^4 + 2*T^3 + 4*T^2 + 10*T + 25)", "verified": "2026-09-18 17:32:21 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "21.c", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "1", "3", "3", "1" ], "h": [ "0", "1", "1", "1" ], "equation": { "f": "x^3 + 3*x^2 + 3*x + 1", "h": "x^3 + x^2 + x", "text": "y^2 + (x^3 + x^2 + x)*y = x^3 + 3*x^2 + 3*x + 1" }, "conductor": "676", "conductor_sort": 676, "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)", "discriminant_digits": 8, "lmfdb": { "label": "676.a.5408.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/676/a/5408/1", "snapshot_label": "" }, "sources": [], "rm": false, "discovered_by": "F. Leprévost", "credit_differs": false, "year": 1995, "historical": false, "first_source": null, "first_source_year": null, "torsion_method": "generators", "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", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-10-14 10:49:22 UTC", "submitted": "2026-10-14", "source_kind": "issue" } ], "gsplit": [ { "id": "21.b", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "0", "0", "1", "2", "2", "1" ], "h": [ "1", "1", "0", "1" ], "equation": { "f": "x^5 + 2*x^4 + 2*x^3 + x^2", "h": "x^3 + x + 1", "text": "y^2 + (x^3 + x + 1)*y = x^5 + 2*x^4 + 2*x^3 + x^2" }, "conductor": "324", "conductor_sort": 324, "conductor_note": "", "discriminant_digits": 7, "lmfdb": { "label": "324.a.648.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/324/a/648/1", "snapshot_label": "" }, "sources": [ "Ogg1973" ], "rm": false, "discovered_by": "Ogg 1973", "credit_differs": false, "year": null, "historical": false, "first_source": "Ogg1973", "first_source_year": 1973, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 19 (source: \\cite{Ogg1973})", "new_group": false, "new_for_class": true, "certificate": "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]; simple over ℚ: χ_5 = T^4 - 5*T^2 + 25 is irreducible", "verified": "2026-09-18 17:36:10 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": 0, "gsplit": null, "undecided": null }, "first_source": { "simple": "Leprevost1991", "qsplit": null, "gsplit": "Ogg1973", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite", "Leprevost1991" ], "note": "Leprévost's 1991 one-parameter degree-5 family f₂₁ = A₂₁² − k₂₁x³(x−1)²: D = [x, A₂₁(0)] has order 21 over ℚ(t). Every 5-integral t ≡ 1 (mod 5) has #J(𝔽₅) = 21, strict (family_21_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite", "HLP2000" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Lemma 11 and Prop. 4) of the Kubert curve E₇(t) (torsion ℤ/7; short model y² = x³ + Ax + B, A = −27c₄, B = −54c₆) with the curve E₃: y² = x³ + (x − B²/A³)² of Lemma 11, which has a point of order 3 and 2-torsion Galois-isomorphic to that of E₇: C_t: y² = A⁷ − 3A⁵Bx² + (A⁶ + 3A³B²)x⁴ − AB³x⁶, with (x, y) ↦ (x², y) onto E₃ and (x, y) ↦ (1/x², y/x³) onto E₇, so J is isogenous over ℚ to E₃ × E₇. The pullbacks of the points of order 7 and 3 mark [21] over ℚ(t). Every 5-integral t ≡ 2 (mod 5) has #J(𝔽₅) = 21, so J(ℚ)_tors is exactly [21]; the G2-invariants of the members differ (family_21_split_certificate.log)", "inherited_from": null } } }, { "key": "22", "group": [ 22 ], "bracket": "[22]", "label": "ℤ/22", "order": 22, "rank": 1, "classes": { "simple": [ { "id": "22.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "1", "-1", "-2", "1" ], "h": [ "0", "1", "0", "1" ], "equation": { "f": "x^3 - 2*x^2 - x + 1", "h": "x^3 + x", "text": "y^2 + (x^3 + x)*y = x^3 - 2*x^2 - x + 1" }, "conductor": "1192", "conductor_sort": 1192, "conductor_note": "", "discriminant_digits": 8, "lmfdb": { "label": "1192.a.19072.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/1192/a/19072/1", "snapshot_label": "" }, "sources": [ "Leprevost1993" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": true, "year": null, "historical": false, "first_source": "Leprevost1993", "first_source_year": 1993, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 22 (source: \\cite{Leprevost1993})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 3 is strict (χ = T^4 + 2*T^3 + 4*T^2 + 6*T + 9)", "verified": "2026-09-18 17:32:22 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "Leprevost1993", "qsplit": null, "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": false, "gsplit": false, "undecided": false }, "exclusions": { "qsplit": { "reason": "A Jacobian split over ℚ has no rational torsion of prime order greater than 7, by Mazur's theorem applied to an elliptic subvariety and its elliptic quotient.", "sources": [ "QSplitPrimeObstruction", "Mazur1977" ], "proof_url": "proofs/qsplit-prime-obstruction.html" }, "gsplit": { "reason": "A geometrically split abelian surface over ℚ has no rational point of order 22. The proof applies to every geometric splitting degree.", "sources": [ "Geometric22Obstruction", "Mazur1977", "KenkuMomose1988", "LeFournNajman2020", "BourdonClark2020" ], "proof_url": "proofs/geometric-22-obstruction.html" } }, "n_curves": 1, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite" ], "note": "Flynn's order-11 family on the curve t = (−s²(s²+1)(s⁴−s²+1) + 2s⁵)/(s²−1)², where x = s² is a root: over ℚ(s), [∞₁ − ∞₂] has order 11 and [(s², 0) − ∞₁] has order 22. Every 5-integral s ≡ 2 (mod 5) has #J(𝔽₅) = 22, strict (family_22_certificate.log)", "inherited_from": null }, "split": { "grade": "impossible", "sources": [ "Geometric22Obstruction", "Mazur1977", "KenkuMomose1988", "LeFournNajman2020", "BourdonClark2020" ], "note": "A geometrically split abelian surface over ℚ has no rational point of order 22. The proof applies to every geometric splitting degree.", "proof_url": "proofs/geometric-22-obstruction.html", "inherited_from": null } } }, { "key": "23", "group": [ 23 ], "bracket": "[23]", "label": "ℤ/23", "order": 23, "rank": 1, "classes": { "simple": [ { "id": "23.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "1", "-2", "2", "1", "-2", "-1", "1" ], "h": [ "0", "0", "1" ], "equation": { "f": "x^6 - x^5 - 2*x^4 + x^3 + 2*x^2 - 2*x + 1", "h": "x^2", "text": "y^2 + x^2*y = x^6 - x^5 - 2*x^4 + x^3 + 2*x^2 - 2*x + 1" }, "conductor": "1696", "conductor_sort": 1696, "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", "discriminant_digits": 10, "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" }, "sources": [ "Leprevost1993" ], "rm": false, "discovered_by": "Leprévost 1993", "credit_differs": false, "year": null, "historical": false, "first_source": "Leprevost1993", "first_source_year": 1993, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 23 (source: \\cite{Leprevost1993})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 3 is strict (χ = T^4 + 2*T^3 + 5*T^2 + 6*T + 9)", "verified": "2026-09-18 17:32:22 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "Leprevost1993", "qsplit": null, "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": false, "gsplit": false, "undecided": false }, "exclusions": { "qsplit": { "reason": "A Jacobian split over ℚ has no rational torsion of prime order greater than 7, by Mazur's theorem applied to an elliptic subvariety and its elliptic quotient.", "sources": [ "QSplitPrimeObstruction", "Mazur1977" ], "proof_url": "proofs/qsplit-prime-obstruction.html" }, "gsplit": { "reason": "A geometrically split abelian surface over ℚ has no rational point of order 23 or 29. The proof applies to every geometric splitting degree.", "sources": [ "Geometric2329Obstruction", "DKSS2023", "Khawaja2024", "Clark2004", "ClarkXarles2008" ], "proof_url": "proofs/geometric-23-29-obstruction.html" } }, "n_curves": 1, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite", "KuruSadek" ], "note": "the Kuru–Sadek quadratic-order construction, genus-2 specialization f_t = A² − λx⁴(x−1), with β = (t²+1)²/(4t²), s = (t²−1)/(2t), α = β − s⁵/β^{3/2}, λ = (α−1)⁴/((α−β)²α): D = [x, A(0)] has order 23 over ℚ(t). Every t ≡ 17 (mod 91) has #J(𝔽₇) = 46 and #J(𝔽₁₃) = 115 (gcd 23), strict at 7 and 13 (family_23_certificate.log)", "inherited_from": null }, "split": { "grade": "impossible", "sources": [ "Geometric2329Obstruction", "DKSS2023", "Khawaja2024", "Clark2004", "ClarkXarles2008" ], "note": "A geometrically split abelian surface over ℚ has no rational point of order 23 or 29. The proof applies to every geometric splitting degree.", "proof_url": "proofs/geometric-23-29-obstruction.html", "inherited_from": null } } }, { "key": "24", "group": [ 24 ], "bracket": "[24]", "label": "ℤ/24", "order": 24, "rank": 1, "classes": { "simple": [ { "id": "24.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "2", "4", "3", "2" ], "h": [ "1", "0", "0", "1" ], "equation": { "f": "2*x^4 + 3*x^3 + 4*x^2 + 2*x", "h": "x^3 + 1", "text": "y^2 + (x^3 + 1)*y = 2*x^4 + 3*x^3 + 4*x^2 + 2*x" }, "conductor": "1908", "conductor_sort": 1908, "conductor_note": "", "discriminant_digits": 9, "lmfdb": { "label": "1908.a.183168.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/1908/a/183168/1", "snapshot_label": "" }, "sources": [ "Leprevost1993" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": true, "year": null, "historical": false, "first_source": "Leprevost1993", "first_source_year": 1993, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 24 (source: \\cite{Leprevost1993})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 17 is strict (χ = T^4 + T^3 - 20*T^2 + 17*T + 289)", "verified": "2026-09-18 17:32:23 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "24.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "375", "0", "97", "0", "8" ], "h": [ "0", "1", "0", "1" ], "equation": { "f": "8*x^4 + 97*x^2 + 375", "h": "x^3 + x", "text": "y^2 + (x^3 + x)*y = 8*x^4 + 97*x^2 + 375" }, "conductor": "300", "conductor_sort": 300, "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)", "discriminant_digits": 11, "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" }, "sources": [ "Leprevost1995" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "Leprevost1995", "first_source_year": 1995, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 20 (source: \\cite{Leprevost1995})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:36:11 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "Leprevost1993", "qsplit": "Leprevost1995", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 2, "infinite": { "simple": { "grade": "exact", "sources": [ "ThisSite" ], "note": "relation-lattice family with lattice (4,1),(0,6): f = A² + λx⁴(x−1) = B² − (x−1)⁶, B − A = (x−1)g₁, B + A = g₂, g₁g₂ = λx⁴ + (x−1)⁵, over the rational curve of quadratics g₁ = x² − e₁x + e₂, e₁ = (−33t⁴+65t³−42t²+11t−1)/t⁴, e₂ = (−9t³+15t²−7t+1)/t³. Every t ≡ 9 (mod 35) has #J(𝔽₅) = 24 and #J(𝔽₇) = 48, strict at 7 (family_24_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite", "HLP2000" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4; the case N = 8, N′ = 6 of §3.5) of E = E₈(3/2) = 21a3 (torsion ℤ/8) with the Kubert curves F_s = E₆(s) on the conic (s + 1)(9s + 1) = 7w², s = (1 − 7m²)/(7m² − 9): C_m: y² = c₂F(c₁x² − 1, c₂x²) with c₁, c₂ ∈ ℚ(m) and F(X, Z) = X³ + pXZ² + qZ³ (E: y² = z³ + pz + q). The maps (x, y) ↦ (1/x², y/x³), (x², y) onto E and F_s make J isogenous over ℚ to E × F_s, and the pullbacks of the points of order 8 and 6 generate ℤ/24 over ℚ(m). Every 5- and 17-integral m ≡ 56 (mod 85) has J(𝔽₅) = ℤ/48 (cyclic 2-part) and J(𝔽₁₇) = [24,24] (2-primary exponent 8), so J(ℚ)_tors is exactly [24]; the G2-invariants of the members differ. The member m = −1 (outside the class) is the census curve 48.a, where the gain of HLP Prop. 12 gives [48] (family_24_split_certificate.log)", "inherited_from": null } } }, { "key": "25", "group": [ 25 ], "bracket": "[25]", "label": "ℤ/25", "order": 25, "rank": 1, "classes": { "simple": [ { "id": "25.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "2", "-23", "41", "75", "25", "-9" ], "h": [ "1", "1" ], "equation": { "f": "-9*x^5 + 25*x^4 + 75*x^3 + 41*x^2 - 23*x + 2", "h": "x + 1", "text": "y^2 + (x + 1)*y = -9*x^5 + 25*x^4 + 75*x^3 + 41*x^2 - 23*x + 2" }, "conductor": "168750", "conductor_sort": 168750, "conductor_note": "", "discriminant_digits": 16, "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" }, "sources": [ "Nicholls2018" ], "rm": false, "discovered_by": "Nicholls 2018", "credit_differs": false, "year": null, "historical": false, "first_source": "Nicholls2018", "first_source_year": 2018, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 25 (source: \\cite{Nicholls2018})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 11 is strict (χ = T^4 - 5*T^3 + 13*T^2 - 55*T + 121)", "verified": "2026-09-18 17:32:31 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [], "gsplit": [ { "id": "25.b", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "1", "8", "13", "9", "14", "4", "2" ], "h": [ "0", "0", "1", "1" ], "equation": { "f": "2*x^6 + 4*x^5 + 14*x^4 + 9*x^3 + 13*x^2 + 8*x + 1", "h": "x^3 + x^2", "text": "y^2 + (x^3 + x^2)*y = 2*x^6 + 4*x^5 + 14*x^4 + 9*x^3 + 13*x^2 + 8*x + 1" }, "conductor": "42336", "conductor_sort": 42336, "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", "discriminant_digits": 17, "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" }, "sources": [ "Leprevost1995", "PZP2013" ], "rm": false, "discovered_by": "Leprévost 1995; Platonov–Zhgun–Petrunin 2013", "credit_differs": false, "year": null, "historical": false, "first_source": "Leprevost1995", "first_source_year": 1995, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 21 (source: \\cite{Leprevost1995,PZP2013})", "new_group": false, "new_for_class": true, "certificate": "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]; simple over ℚ: χ_5 = T^4 - T^2 + 25 is irreducible", "verified": "2026-09-18 17:36:19 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "Nicholls2018", "qsplit": null, "gsplit": "Leprevost1995", "undecided": null }, "known": { "simple": true, "qsplit": false, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 2, "infinite": { "simple": { "grade": "exact", "sources": [ "ThisSite" ], "note": "relation-lattice family with lattice (5,0),(1,5): f = A² + λx⁵ = B² − x(x−1)⁵, B − A = xg₁, B + A = g₂, over the same rational curve of quadratics g₁ as for [24]. Every t ≡ 3 (mod 77) has #J(𝔽₇) = 50 and #J(𝔽₁₁) = 75, strict at 11 (family_25_certificate.log)", "inherited_from": null }, "split": { "grade": "open", "sources": [], "note": "", "inherited_from": null } } }, { "key": "26", "group": [ 26 ], "bracket": "[26]", "label": "ℤ/26", "order": 26, "rank": 1, "classes": { "simple": [ { "id": "26.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "0", "-3", "-3", "-3", "-12" ], "h": [ "1", "1" ], "equation": { "f": "-12*x^5 - 3*x^4 - 3*x^3 - 3*x^2", "h": "x + 1", "text": "y^2 + (x + 1)*y = -12*x^5 - 3*x^4 - 3*x^3 - 3*x^2" }, "conductor": "11016", "conductor_sort": 11016, "conductor_note": "", "discriminant_digits": 11, "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" }, "sources": [ "Leprevost1993" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "Leprevost1993", "first_source_year": 1993, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 26 (source: \\cite{Leprevost1993})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 11 is strict (χ = T^4 + 4*T^3 + 12*T^2 + 44*T + 121)", "verified": "2026-09-18 17:32:32 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "Leprevost1993", "qsplit": null, "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": false, "gsplit": false, "undecided": false }, "exclusions": { "qsplit": { "reason": "A Jacobian split over ℚ has no rational torsion of prime order greater than 7, by Mazur's theorem applied to an elliptic subvariety and its elliptic quotient.", "sources": [ "QSplitPrimeObstruction", "Mazur1977" ], "proof_url": "proofs/qsplit-prime-obstruction.html" }, "gsplit": { "reason": "A geometrically split abelian surface over ℚ has no rational point of order 26 or 39. The proof applies to every geometric splitting degree.", "sources": [ "Geometric2639Obstruction", "Mazur1977", "KenkuMomose1988", "DEvHMZB2021", "DNQuartic2025", "Faltings1983", "SerreTate1968", "Clark2004", "ClarkXarles2008" ], "proof_url": "proofs/geometric-26-39-obstruction.html" } }, "n_curves": 1, "infinite": { "simple": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the rational curve s = (16/3)(1+3t)⁴/d(t), d(t) = 1 + 28t + 312t² + 1728t³ + 4752t⁴ + 5184t⁵, on the [13] family, where f_s has the rational root x₀ = −(1/3)(1+6t)⁶/d(t): Z/26 is marked over ℚ(t). Every t ≡ 17 (mod 35) has #J(𝔽₅) = 26 and #J(𝔽₇) = 78, strict at 7 (family_26_certificate.log)", "inherited_from": null }, "split": { "grade": "impossible", "sources": [ "Geometric2639Obstruction", "Mazur1977", "KenkuMomose1988", "DEvHMZB2021", "DNQuartic2025", "Faltings1983", "SerreTate1968", "Clark2004", "ClarkXarles2008" ], "note": "A geometrically split abelian surface over ℚ has no rational point of order 26 or 39. The proof applies to every geometric splitting degree.", "proof_url": "proofs/geometric-26-39-obstruction.html", "inherited_from": null } } }, { "key": "27", "group": [ 27 ], "bracket": "[27]", "label": "ℤ/27", "order": 27, "rank": 1, "classes": { "simple": [ { "id": "27.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "-1", "1", "1", "-1" ], "h": [ "1", "0", "0", "1" ], "equation": { "f": "-x^4 + x^3 + x^2 - x", "h": "x^3 + 1", "text": "y^2 + (x^3 + 1)*y = -x^4 + x^3 + x^2 - x" }, "conductor": "604", "conductor_sort": 604, "conductor_note": "", "discriminant_digits": 8, "lmfdb": { "label": "604.a.9664.2", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/604/a/9664/2", "snapshot_label": "" }, "sources": [ "Howe2015" ], "rm": false, "discovered_by": "Howe 2015", "credit_differs": false, "year": null, "historical": false, "first_source": "Howe2015", "first_source_year": 2015, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 27 (source: \\cite{Howe2015})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 3 is strict (χ = T^4 + 3*T^3 + 5*T^2 + 9*T + 9)", "verified": "2026-09-18 17:32:33 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "27.c", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "9", "-36", "90", "-192", "249", "-60", "4" ], "h": [ "0" ], "equation": { "f": "4*x^6 - 60*x^5 + 249*x^4 - 192*x^3 + 90*x^2 - 36*x + 9", "h": "0", "text": "y^2 = 4*x^6 - 60*x^5 + 249*x^4 - 192*x^3 + 90*x^2 - 36*x + 9" }, "conductor": "8748", "conductor_sort": 8748, "conductor_note": "", "discriminant_digits": 23, "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": "" }, "sources": [], "rm": false, "discovered_by": "F. Leprévost", "credit_differs": false, "year": 1995, "historical": false, "first_source": null, "first_source_year": null, "torsion_method": "generators", "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.", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-10-14 10:49:32 UTC", "submitted": "2026-10-14", "source_kind": "issue" } ], "gsplit": [ { "id": "27.b", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "0", "4", "20", "17", "9", "0", "1" ], "h": [ "1", "1" ], "equation": { "f": "x^6 + 9*x^4 + 17*x^3 + 20*x^2 + 4*x", "h": "x + 1", "text": "y^2 + (x + 1)*y = x^6 + 9*x^4 + 17*x^3 + 20*x^2 + 4*x" }, "conductor": "7776", "conductor_sort": 7776, "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", "discriminant_digits": 16, "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" }, "sources": [ "Howe2015" ], "rm": false, "discovered_by": "Howe 2015", "credit_differs": false, "year": null, "historical": false, "first_source": "Howe2015", "first_source_year": 2015, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 22 (source: \\cite{Leprevost1995})", "new_group": false, "new_for_class": true, "certificate": "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]; simple over ℚ: χ_7 = T^4 + 4*T^2 + 49 is irreducible", "verified": "2026-09-18 17:37:17 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": 0, "gsplit": null, "undecided": null }, "first_source": { "simple": "Howe2015", "qsplit": null, "gsplit": "Howe2015", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings, in the normal form of Bröker–Howe–Lauter–Stevenhagen (arXiv:1403.6911), of E = 54b3 (torsion ℤ/9, P₉ = (9,19)) with the partners F_w anti-symplectically 3-congruent to E (the family of the split [9] record). Here the class of order 27 is a 3-power gain of the gluing, not a pullback: with π = φ₁^* + φ₂^* and π̂ = (φ₁_*, φ₂_*), the rational points x of the fibre π̂^{-1}(P₉ ± P₃, ±T) (T the rational 3-torsion point of F_w) satisfy 3x = φ₁^*(P₉). Translation by φ₁^*(P₃) acts freely on this genus-1 locus (a twist of the modular curve of Γ(3) ∩ Γ₁(9); its untwisted form {Q ∈ F_w : 3Q = T} is computed to have genus 1), and its quotient is the cube curve C: z³ = 4g(w), g = 292w³ + 207w² − 378w + 135, read off from the 3-isogeny descent classes (y(P₉) = 4 on E, A₃(F_w) = 2g(w) mod cubes). Jac(C): y² = x³ − 972 has rank 1, and the cover is the 3-isogeny from y² = x³ + 36. At R₀ = (−3/2, 9) and S = (3/7, 36/7) the Jacobian has J(ℚ)_tors = ℤ/27, and both classes lie on the component over C; the locus is an étale triple cover of C, so with a point over R₀ as origin it is an isogeny onto (C, R₀), the points of C(ℚ) that lift form a coset R₀ + H, and every R_k = R₀ + 2k(S − R₀) lifts, i.e. carries a rational class of order 27. S − R₀ has order 2 modulo 7, so w(R_k) ≡ 2 (mod 7), where J(𝔽₇) ≅ ℤ/27; hence J(ℚ)_tors is exactly [27]. The G2-invariants at k = 0, ±1 differ (family_27_split_certificate.log)", "inherited_from": null } } }, { "key": "28", "group": [ 28 ], "bracket": "[28]", "label": "ℤ/28", "order": 28, "rank": 1, "classes": { "simple": [ { "id": "28.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "2", "3", "1", "1", "0", "-1" ], "h": [ "1", "0", "0", "1" ], "equation": { "f": "-x^5 + x^3 + x^2 + 3*x + 2", "h": "x^3 + 1", "text": "y^2 + (x^3 + 1)*y = -x^5 + x^3 + x^2 + 3*x + 2" }, "conductor": "249", "conductor_sort": 249, "conductor_note": "", "discriminant_digits": 8, "lmfdb": { "label": "249.a.6723.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/249/a/6723/1", "snapshot_label": "" }, "sources": [ "PP2012", "PZP2013" ], "rm": false, "discovered_by": "Platonov–Petrunin 2012a; Platonov–Zhgun–Petrunin 2013", "credit_differs": false, "year": null, "historical": false, "first_source": "PP2012", "first_source_year": 2012, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 28 (source: \\cite{PP2012,PZP2013})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 7 is strict (χ = T^4 + T^3 - 2*T^2 + 7*T + 49)", "verified": "2026-09-18 17:32:33 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "28.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "4", "10", "-4", "-1", "5", "-3", "1" ], "h": [ "0", "1", "1" ], "equation": { "f": "x^6 - 3*x^5 + 5*x^4 - x^3 - 4*x^2 + 10*x + 4", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = x^6 - 3*x^5 + 5*x^4 - x^3 - 4*x^2 + 10*x + 4" }, "conductor": "4732", "conductor_sort": 4732, "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", "discriminant_digits": 15, "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" }, "sources": [ "PZP2013", "Howe2015" ], "rm": false, "discovered_by": "Platonov–Zhgun–Petrunin 2013; Howe 2015", "credit_differs": false, "year": null, "historical": false, "first_source": "PZP2013", "first_source_year": 2013, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 23 (source: \\cite{PZP2013,Howe2015})", "new_group": false, "new_for_class": true, "certificate": "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]", "verified": "2026-09-18 17:37:19 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "PP2012", "qsplit": "PZP2013", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 2, "infinite": { "simple": { "grade": "exact", "sources": [ "PlatonovFedorov2018", "ThisSite" ], "note": "Platonov–Fedorov 2018, formula (8): C_t: y² = a₅x⁵ + a₄x⁴ + a₃x³ + a₂x² + a₁x + (t + 1)⁴(t² + t + 1)², a₅ = −4t⁵(t + 1)(2t + 1)(t² + t + 1)² (a₄, …, a₁ as in (8) and in family_28_certificate.m), built from ω₁² − f = (x − 1)⁵x and ω₃² − ω₄²f = (x − 1)²x⁶, i.e. functions with divisors 5P + Q − 6∞ and 2P + 6Q − 8∞. Over ℚ(t) the class D = [x − 1, −t⁵(t + 1)] = P − ∞ has order 28 (14D = [x − (t + 1)²/(2t + 1), 0], Q − ∞ = −5D). Every 5- and 7-integral t ≡ 1 (mod 35) has #J(𝔽₅) = 28 and #J(𝔽₇) = 56 (gcd 28), strict at 7 (family_28_certificate.log). The census curves 28.a and 28.b are not on the family (28.b is a fiber only over ℚ(√−7))", "inherited_from": null }, "split": { "grade": "open", "sources": [], "note": "", "inherited_from": null } } }, { "key": "29", "group": [ 29 ], "bracket": "[29]", "label": "ℤ/29", "order": 29, "rank": 1, "classes": { "simple": [ { "id": "29.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "0", "-1", "2", "0", "-2", "1" ], "h": [ "1", "1" ], "equation": { "f": "x^6 - 2*x^5 + 2*x^3 - x^2", "h": "x + 1", "text": "y^2 + (x + 1)*y = x^6 - 2*x^5 + 2*x^3 - x^2" }, "conductor": "976", "conductor_sort": 976, "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", "discriminant_digits": 10, "lmfdb": { "label": "976.a.999424.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/976/a/999424/1", "snapshot_label": "" }, "sources": [ "Leprevost1993" ], "rm": false, "discovered_by": "Leprévost 1993", "credit_differs": false, "year": null, "historical": false, "first_source": "Leprevost1993", "first_source_year": 1993, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 29 (source: \\cite{Leprevost1993})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 3 is strict (χ = T^4 + 3*T^3 + 7*T^2 + 9*T + 9)", "verified": "2026-09-18 17:32:34 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "Leprevost1993", "qsplit": null, "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": false, "gsplit": false, "undecided": false }, "exclusions": { "qsplit": { "reason": "A Jacobian split over ℚ has no rational torsion of prime order greater than 7, by Mazur's theorem applied to an elliptic subvariety and its elliptic quotient.", "sources": [ "QSplitPrimeObstruction", "Mazur1977" ], "proof_url": "proofs/qsplit-prime-obstruction.html" }, "gsplit": { "reason": "A geometrically split abelian surface over ℚ has no rational point of order 23 or 29. The proof applies to every geometric splitting degree.", "sources": [ "Geometric2329Obstruction", "DKSS2023", "Khawaja2024", "Clark2004", "ClarkXarles2008" ], "proof_url": "proofs/geometric-23-29-obstruction.html" } }, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "impossible", "sources": [ "Geometric2329Obstruction", "DKSS2023", "Khawaja2024", "Clark2004", "ClarkXarles2008" ], "note": "A geometrically split abelian surface over ℚ has no rational point of order 23 or 29. The proof applies to every geometric splitting degree.", "proof_url": "proofs/geometric-23-29-obstruction.html", "inherited_from": null } } }, { "key": "30", "group": [ 30 ], "bracket": "[30]", "label": "ℤ/30", "order": 30, "rank": 1, "classes": { "simple": [ { "id": "30.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "-1", "16", "-12", "-10", "4", "2" ], "h": [ "0", "0", "1", "1" ], "equation": { "f": "2*x^6 + 4*x^5 - 10*x^4 - 12*x^3 + 16*x^2 - x", "h": "x^3 + x^2", "text": "y^2 + (x^3 + x^2)*y = 2*x^6 + 4*x^5 - 10*x^4 - 12*x^3 + 16*x^2 - x" }, "conductor": "8694", "conductor_sort": 8694, "conductor_note": "", "discriminant_digits": 13, "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" }, "sources": [ "PP2015" ], "rm": false, "discovered_by": "Platonov–Petrunin 2015", "credit_differs": false, "year": null, "historical": false, "first_source": "PP2015", "first_source_year": 2015, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 30 (source: \\cite{PP2015})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 13 is strict (χ = T^4 - 2*T^3 + 8*T^2 - 26*T + 169)", "verified": "2026-09-18 17:32:35 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "30.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "2", "6", "10", "11", "10", "6", "2" ], "h": [ "1", "0", "0", "1" ], "equation": { "f": "2*x^6 + 6*x^5 + 10*x^4 + 11*x^3 + 10*x^2 + 6*x + 2", "h": "x^3 + 1", "text": "y^2 + (x^3 + 1)*y = 2*x^6 + 6*x^5 + 10*x^4 + 11*x^3 + 10*x^2 + 6*x + 2" }, "conductor": "4356", "conductor_sort": 4356, "conductor_note": "N(E1) N(E2) = 66 * 66 for the elliptic curves ['1', '0', '0', '-45', '81'] and ['1', '0', '1', '-6', '4'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "discriminant_digits": 12, "lmfdb": { "label": "4356.j.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B2%2C6%2C10%2C11%2C10%2C6%2C2%5D%2C%5B1%2C0%2C0%2C1%5D%5D", "snapshot_label": "4356.j.1" }, "sources": [ "HLP2000" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "HLP2000", "first_source_year": 2000, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 24 (source: \\cite{HLP2000})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:37:20 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "PP2015", "qsplit": "HLP2000", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": { "gsplit": { "reason": "A geometrically split abelian surface that is simple over ℚ has no rational point of order 30. The proof applies to every geometric splitting degree and requires simplicity over ℚ.", "sources": [ "QSimple30Obstruction", "DNQuartic2025", "SerreTate1968", "Tate1966", "Mazur1977", "KenkuMomose1988", "ClarkXarles2008", "Clark2004" ], "proof_url": "proofs/qsimple-30-obstruction.html" } }, "n_curves": 2, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite" ], "note": "the simultaneous contact-5 / contact-6 family f = h₆² − (x−1)⁶ = h₅² − Kx⁵ over ℚ(R), branch −1: D₅ = [x, h₅(0)] has order 5 and D₆ = [x − 1, h₆(1)] has order 6. Every R ≡ 110 (mod 143) has #J(𝔽₁₁) = 150 and #J(𝔽₁₃) = 120 (gcd 30), strict at 13 (family_30_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite", "HLP2000" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4) of E₁ = 66c1 (torsion ℤ/10, non-rational 2-torsion over ℚ(√33)) with the Kubert curve E₂ = E₆(u), u = (33s² − 1)/(9 − 33s²) (the line through u = −1 on the conic Δ₁₀w² = (u + 1)(9u + 1) of HLP 2000, §3.5), along the Galois isomorphism of 2-torsion fixing the rational points. Up to scaling the HLP sextic is C_s: y² = F(x²), with (x, y) ↦ (x², y) onto E₂ and (x, y) ↦ (1/x², y/x³) onto E₁, so J is isogenous over ℚ to E₁ × E₂. Over ℚ(s) the pullbacks of the points of order 10 and 6 mark [30]. Here #J(𝔽_p) is always divisible by 60, so the group structure is used: every 7- and 13-integral s ≡ 1 (mod 91) has J(𝔽₇) ≅ [2,30] and J(𝔽₁₃) ≅ ℤ/120 (cyclic), so J(ℚ)_tors is exactly [30]; the G2-invariants of the members differ. The census curve 30.b is the member s = −1/3 (family_30_split_certificate.log)", "inherited_from": null } } }, { "key": "31", "group": [ 31 ], "bracket": "[31]", "label": "ℤ/31", "order": 31, "rank": 1, "classes": { "simple": [ { "id": "31.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "-126", "816", "-2466", "4300", "-4587", "2841", "-839" ], "h": [ "0", "-1", "-1" ], "equation": { "f": "-839*x^6 + 2841*x^5 - 4587*x^4 + 4300*x^3 - 2466*x^2 + 816*x - 126", "h": "-x^2 - x", "text": "y^2 + (-x^2 - x)*y = -839*x^6 + 2841*x^5 - 4587*x^4 + 4300*x^3 - 2466*x^2 + 816*x - 126" }, "conductor": "3348900", "conductor_sort": 3348900, "conductor_note": "", "discriminant_digits": 26, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-126%2C816%2C-2466%2C4300%2C-4587%2C2841%2C-839%5D%2C%5B0%2C-1%2C-1%5D%5D", "snapshot_label": "" }, "sources": [ "costa" ], "rm": true, "discovered_by": "Costa et al. (ModularAbelianSurfaces)", "credit_differs": false, "year": null, "historical": false, "first_source": "costa", "first_source_year": 2022, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 31 (source: \\cite{costa}$^{\\text{RM}}$)", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 11 is strict (χ = T^4 - 8*T^3 + 36*T^2 - 88*T + 121)", "verified": "2026-09-18 17:32:36 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "costa", "qsplit": null, "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": false, "gsplit": false, "undecided": false }, "exclusions": { "qsplit": { "reason": "A Jacobian split over ℚ has no rational torsion of prime order greater than 7, by Mazur's theorem applied to an elliptic subvariety and its elliptic quotient.", "sources": [ "QSplitPrimeObstruction", "Mazur1977" ], "proof_url": "proofs/qsplit-prime-obstruction.html" }, "gsplit": { "reason": "A geometrically split abelian surface over ℚ has no rational point of order 31. The proof applies to every geometric splitting degree.", "sources": [ "Geometric31Obstruction", "DKSS2023", "Khawaja2024", "Clark2004", "ClarkXarles2008" ], "proof_url": "proofs/geometric-31-obstruction.html" } }, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [ "BNSS2026", "Najman2026" ], "note": "every known example has real multiplication (by ℤ[√2]); a single curve is known", "inherited_from": null }, "split": { "grade": "impossible", "sources": [ "Geometric31Obstruction", "DKSS2023", "Khawaja2024", "Clark2004", "ClarkXarles2008" ], "note": "A geometrically split abelian surface over ℚ has no rational point of order 31. The proof applies to every geometric splitting degree.", "proof_url": "proofs/geometric-31-obstruction.html", "inherited_from": null } } }, { "key": "32", "group": [ 32 ], "bracket": "[32]", "label": "ℤ/32", "order": 32, "rank": 1, "classes": { "simple": [ { "id": "32.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "-47", "-49", "49", "45", "1", "-1", "1" ], "h": [ "0", "1", "1" ], "equation": { "f": "x^6 - x^5 + x^4 + 45*x^3 + 49*x^2 - 49*x - 47", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = x^6 - x^5 + x^4 + 45*x^3 + 49*x^2 - 49*x - 47" }, "conductor": "66852", "conductor_sort": 66852, "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", "discriminant_digits": 20, "lmfdb": { "label": "66852.b.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B-47%2C-49%2C49%2C45%2C1%2C-1%2C1%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "66852.b.1" }, "sources": [ "Elkies2002" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "Elkies2002", "first_source_year": 2002, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 32 (source: \\cite{Elkies2002})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 7 is strict (χ = T^4 + T^3 + 6*T^2 + 7*T + 49)", "verified": "2026-09-18 17:32:36 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "Elkies2002", "qsplit": null, "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": false, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 1, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite", "Elkies2002" ], "note": "the reconstructed Elkies order-32 component, parametrized as y² = h² − a²x⁵(x+1), h = ax³ + bx² + cx + 1, a = −2w³(w⁴ − w³ + 2w² − w + 1), r = 1/(w⁴ − 1) (b, c in the certificate): over ℚ(w), D = [x, 1] has 16D = [x − r, 0] ≠ 0 and 32D = 0. The member w = 2 is Elkies' printed curve. Every w ≡ 16 (mod 91) has #J(𝔽₇) = 64 and #J(𝔽₁₃) = 160 (gcd 32), strict at 7 (family_32_certificate.log)", "inherited_from": null }, "split": { "grade": "open", "sources": [], "note": "", "inherited_from": null } } }, { "key": "33", "group": [ 33 ], "bracket": "[33]", "label": "ℤ/33", "order": 33, "rank": 1, "classes": { "simple": [ { "id": "33.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "2", "13", "21", "7", "9", "-7", "1" ], "h": [ "1", "1", "1" ], "equation": { "f": "x^6 - 7*x^5 + 9*x^4 + 7*x^3 + 21*x^2 + 13*x + 2", "h": "x^2 + x + 1", "text": "y^2 + (x^2 + x + 1)*y = x^6 - 7*x^5 + 9*x^4 + 7*x^3 + 21*x^2 + 13*x + 2" }, "conductor": "13716", "conductor_sort": 13716, "conductor_note": "", "discriminant_digits": 16, "lmfdb": { "label": "13716.a.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B2%2C13%2C21%2C7%2C9%2C-7%2C1%5D%2C%5B1%2C1%2C1%5D%5D", "snapshot_label": "13716.a.1" }, "sources": [ "PP2012b", "PZP2013" ], "rm": false, "discovered_by": "Platonov–Petrunin 2012b; Platonov–Zhgun–Petrunin 2013", "credit_differs": false, "year": null, "historical": false, "first_source": "PP2012b", "first_source_year": 2012, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 33 (source: \\cite{PP2012b,PZP2013})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 13 is strict (χ = T^4 - 2*T^3 - 10*T^2 - 26*T + 169)", "verified": "2026-09-18 17:32:37 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "PP2012b", "qsplit": null, "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": false, "gsplit": false, "undecided": false }, "exclusions": { "qsplit": { "reason": "A Jacobian split over ℚ has no rational torsion of prime order greater than 7, by Mazur's theorem applied to an elliptic subvariety and its elliptic quotient.", "sources": [ "QSplitPrimeObstruction", "Mazur1977" ], "proof_url": "proofs/qsplit-prime-obstruction.html" }, "gsplit": { "reason": "A geometrically split abelian surface over ℚ has no rational point of order 33 or 34. The proof applies to every geometric splitting degree.", "sources": [ "Geometric3334Obstruction", "Mazur1977", "DNQuartic2025", "Clark2004", "ClarkXarles2008" ], "proof_url": "proofs/geometric-33-34-obstruction.html" } }, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "impossible", "sources": [ "Geometric3334Obstruction", "Mazur1977", "DNQuartic2025", "Clark2004", "ClarkXarles2008" ], "note": "A geometrically split abelian surface over ℚ has no rational point of order 33 or 34. The proof applies to every geometric splitting degree.", "proof_url": "proofs/geometric-33-34-obstruction.html", "inherited_from": null } } }, { "key": "34", "group": [ 34 ], "bracket": "[34]", "label": "ℤ/34", "order": 34, "rank": 1, "classes": { "simple": [ { "id": "34.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "15", "12", "7", "3", "0", "-2" ], "h": [ "0", "0", "1", "1" ], "equation": { "f": "-2*x^5 + 3*x^3 + 7*x^2 + 12*x + 15", "h": "x^3 + x^2", "text": "y^2 + (x^3 + x^2)*y = -2*x^5 + 3*x^3 + 7*x^2 + 12*x + 15" }, "conductor": "17856", "conductor_sort": 17856, "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", "discriminant_digits": 15, "lmfdb": { "label": "17856.m.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B15%2C12%2C7%2C3%2C0%2C-2%5D%2C%5B0%2C0%2C1%2C1%5D%5D", "snapshot_label": "17856.m.1" }, "sources": [ "Elkies2002" ], "rm": false, "discovered_by": "Elkies 2002", "credit_differs": false, "year": null, "historical": false, "first_source": "Elkies2002", "first_source_year": 2002, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 34 (source: \\cite{Elkies2002})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 5 is strict (χ = T^4 + T^3 + 2*T^2 + 5*T + 25)", "verified": "2026-09-18 17:32:38 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "Elkies2002", "qsplit": null, "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": false, "gsplit": false, "undecided": false }, "exclusions": { "qsplit": { "reason": "A Jacobian split over ℚ has no rational torsion of prime order greater than 7, by Mazur's theorem applied to an elliptic subvariety and its elliptic quotient.", "sources": [ "QSplitPrimeObstruction", "Mazur1977" ], "proof_url": "proofs/qsplit-prime-obstruction.html" }, "gsplit": { "reason": "A geometrically split abelian surface over ℚ has no rational point of order 33 or 34. The proof applies to every geometric splitting degree.", "sources": [ "Geometric3334Obstruction", "Mazur1977", "DNQuartic2025", "Clark2004", "ClarkXarles2008" ], "proof_url": "proofs/geometric-33-34-obstruction.html" } }, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "impossible", "sources": [ "Geometric3334Obstruction", "Mazur1977", "DNQuartic2025", "Clark2004", "ClarkXarles2008" ], "note": "A geometrically split abelian surface over ℚ has no rational point of order 33 or 34. The proof applies to every geometric splitting degree.", "proof_url": "proofs/geometric-33-34-obstruction.html", "inherited_from": null } } }, { "key": "35", "group": [ 35 ], "bracket": "[35]", "label": "ℤ/35", "order": 35, "rank": 1, "classes": { "simple": [], "qsplit": [ { "id": "35.a", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "160", "80", "166", "182", "116", "30", "10" ], "h": [ "0", "1", "1" ], "equation": { "f": "10*x^6 + 30*x^5 + 116*x^4 + 182*x^3 + 166*x^2 + 80*x + 160", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = 10*x^6 + 30*x^5 + 116*x^4 + 182*x^3 + 166*x^2 + 80*x + 160" }, "conductor": "7436", "conductor_sort": 7436, "conductor_note": "N(E1) N(E2) = 286 * 26 for the elliptic curves ['1', '1', '1', '280', '393'] and ['1', '-1', '1', '-3', '3'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "discriminant_digits": 28, "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" }, "sources": [ "HLP2000" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "HLP2000", "first_source_year": 2000, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 25 (source: \\cite{HLP2000})", "new_group": true, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:37:22 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": null, "qsplit": "HLP2000", "gsplit": null, "undecided": null }, "known": { "simple": false, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": { "gsplit": { "reason": "A geometrically split abelian surface that is simple over ℚ has no rational point of order 35. The proof applies to every geometric splitting degree and requires simplicity over ℚ.", "sources": [ "QSimple35Obstruction", "DNQuartic2025", "Clark2004", "ClarkXarles2008" ], "proof_url": "proofs/qsimple-35-obstruction.html" } }, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite", "HLP2000" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4, §3.6) of E = E₇(−1) = 26b1 (torsion ℤ/7) with the Kubert curves F_t = E₅(t) along a Galois-module isomorphism F_t[2] → E[2]. The pairs (t, ψ) form the curve X′ that HLP show is 3-isogenous to y² = −26t(t² − 11t − 1); here X′ = EM: y² = x³ + x² − 901x − 7461 (conductor 54080, rank 2), and ψ is given by a Tschirnhausen transformation φ(z) = c₁z + c₂(z² + 2p/3) with c₁, c₂ ∈ ℚ(EM). In closed form the gluing is C_R: y² = c₂F(c₁x² − 1, c₂x²), F(X, Z) = X³ + pXZ² + qZ³ (E: y² = z³ + pz + q); (x, y) ↦ (1/x², y/x³) and (x, y) ↦ (x², y) map C_R onto E and F_t, so J is isogenous over ℚ to E × F_t, and the pullbacks are injective, so ℤ/35 ⊂ J(ℚ). For R = (1 + 3k)P, P = (91, −820) (t = 32/13), the curve has the reduction of C_P at 3, where #J(𝔽₃) = 35 (forced: #E(𝔽₃) = 7, #F_t(𝔽₃) = 5), so J(ℚ)_tors is exactly [35]; the G2-invariants at P and 2P differ. The census curve 35.a is the member at (−25, 8) (family_35_split_certificate.log)", "inherited_from": null } } }, { "key": "36", "group": [ 36 ], "bracket": "[36]", "label": "ℤ/36", "order": 36, "rank": 1, "classes": { "simple": [ { "id": "36.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "9", "6", "-14", "13", "30", "3" ], "h": [ "0", "0", "1" ], "equation": { "f": "3*x^5 + 30*x^4 + 13*x^3 - 14*x^2 + 6*x + 9", "h": "x^2", "text": "y^2 + x^2*y = 3*x^5 + 30*x^4 + 13*x^3 - 14*x^2 + 6*x + 9" }, "conductor": "59040", "conductor_sort": 59040, "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", "discriminant_digits": 16, "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" }, "sources": [ "PP2015" ], "rm": false, "discovered_by": "Platonov–Petrunin 2015", "credit_differs": false, "year": null, "historical": false, "first_source": "PP2015", "first_source_year": 2015, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 35 (source: \\cite{PP2015})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 11 is strict (χ = T^4 - 2*T^3 + 10*T^2 - 22*T + 121)", "verified": "2026-09-18 17:32:39 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "36.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "9", "-9", "-1", "-5", "5", "-1", "1" ], "h": [ "0", "1", "1" ], "equation": { "f": "x^6 - x^5 + 5*x^4 - 5*x^3 - x^2 - 9*x + 9", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = x^6 - x^5 + 5*x^4 - 5*x^3 - x^2 - 9*x + 9" }, "conductor": "4860", "conductor_sort": 4860, "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", "discriminant_digits": 15, "lmfdb": { "label": "4860.f.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B9%2C-9%2C-1%2C-5%2C5%2C-1%2C1%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "4860.f.1" }, "sources": [ "PP2012b", "Platonov2014" ], "rm": false, "discovered_by": "Platonov–Zhgun–Petrunin 2013", "credit_differs": true, "year": null, "historical": false, "first_source": "PP2012b", "first_source_year": 2012, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 26 (source: \\cite{PP2012b,Platonov2014})", "new_group": false, "new_for_class": true, "certificate": "a nonconstant map of degree 3 from y^2 = g to the genus-1 curve v^2 = (1)u^3 + (0)u^2 + (-157707)u + (78888006) over Q (the elliptic curve 90c3), (x,y) -> (p/q, y*hh/q^2) with p = -549*x^3 + 1647/2*x^2 + 1890*x - 1890, q = x^3 - 3/2*x^2 + 6*x - 6, hh = 31104*x^2 - 69984*x + 46656; degree-3 map to the elliptic curve 90c3 over Q [Genus2Torsion/verify_split_certificates.m, cover 8]", "verified": "2026-09-18 17:37:23 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "PP2015", "qsplit": "PP2012b", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 2, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings, in the normal form of Bröker–Howe–Lauter–Stevenhagen (arXiv:1403.6911), of E = 90c3 (torsion ℤ/12) with the partners F_w anti-symplectically 3-congruent to E (the family of the split [12] record), on the locus where F_w has a rational point Q of order 9 with 3Q = T, T the rational 3-torsion point of F_w. Then φ₁^*(ℤ/12) and φ₂^*(Q) generate ℤ/36 in J(ℚ) (pullbacks along maps of prime degree are injective, and 3φ₂^*(Q) = −φ₁^*(P₃)). The locus {Q ∈ F_w : 3Q = T} is irreducible of degree 9 over ℚ(w) and of genus 1. Its quotient by Q ↦ Q + T is the cube curve C: z³ = 25g(w), g = 890428w³ + 1234359w² + 336798w + 98415, whose Jacobian y² + y = x³ − 1 has rank 1. The points R₀ = (0, 135) and S = (−81/119, 2700/17) lift (the latter is the census curve 36.b); the locus is an étale triple cover of C, so the points of C(ℚ) that lift form a coset R₀ + H (an isogeny once a point over R₀ is the origin), and every R_k = R₀ + 4k(S − R₀) lifts. Then w(R_k) ≡ 0 modulo 11 and 37, where J(𝔽₁₁) ≅ ℤ/108 and J(𝔽₃₇) ≅ [36,36]; the largest group embedding in both is [36], so J(ℚ)_tors is exactly [36]. The G2-invariants at k = 0, 1 differ (family_36_split_certificate.log)", "inherited_from": null } } }, { "key": "39", "group": [ 39 ], "bracket": "[39]", "label": "ℤ/39", "order": 39, "rank": 1, "classes": { "simple": [ { "id": "39.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "-1", "1", "2", "1" ], "h": [ "1", "0", "0", "1" ], "equation": { "f": "x^4 + 2*x^3 + x^2 - x", "h": "x^3 + 1", "text": "y^2 + (x^3 + 1)*y = x^4 + 2*x^3 + x^2 - x" }, "conductor": "1116", "conductor_sort": 1116, "conductor_note": "", "discriminant_digits": 9, "lmfdb": { "label": "1116.a.214272.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/1116/a/214272/1", "snapshot_label": "" }, "sources": [ "Elkies2002" ], "rm": false, "discovered_by": "Elkies 2002", "credit_differs": false, "year": null, "historical": false, "first_source": "Elkies2002", "first_source_year": 2002, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 36 (source: \\cite{Elkies2002})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 5 is strict (χ = T^4 + 2*T^3 + T^2 + 10*T + 25)", "verified": "2026-09-18 17:32:40 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "Elkies2002", "qsplit": null, "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": false, "gsplit": false, "undecided": false }, "exclusions": { "qsplit": { "reason": "A Jacobian split over ℚ has no rational torsion of prime order greater than 7, by Mazur's theorem applied to an elliptic subvariety and its elliptic quotient.", "sources": [ "QSplitPrimeObstruction", "Mazur1977" ], "proof_url": "proofs/qsplit-prime-obstruction.html" }, "gsplit": { "reason": "A geometrically split abelian surface over ℚ has no rational point of order 26 or 39. The proof applies to every geometric splitting degree.", "sources": [ "Geometric2639Obstruction", "Mazur1977", "KenkuMomose1988", "DEvHMZB2021", "DNQuartic2025", "Faltings1983", "SerreTate1968", "Clark2004", "ClarkXarles2008" ], "proof_url": "proofs/geometric-26-39-obstruction.html" } }, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "impossible", "sources": [ "Geometric2639Obstruction", "Mazur1977", "KenkuMomose1988", "DEvHMZB2021", "DNQuartic2025", "Faltings1983", "SerreTate1968", "Clark2004", "ClarkXarles2008" ], "note": "A geometrically split abelian surface over ℚ has no rational point of order 26 or 39. The proof applies to every geometric splitting degree.", "proof_url": "proofs/geometric-26-39-obstruction.html", "inherited_from": null } } }, { "key": "40", "group": [ 40 ], "bracket": "[40]", "label": "ℤ/40", "order": 40, "rank": 1, "classes": { "simple": [ { "id": "40.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "0", "-2", "-2", "4", "3" ], "h": [ "1", "1" ], "equation": { "f": "3*x^5 + 4*x^4 - 2*x^3 - 2*x^2", "h": "x + 1", "text": "y^2 + (x + 1)*y = 3*x^5 + 4*x^4 - 2*x^3 - 2*x^2" }, "conductor": "1206", "conductor_sort": 1206, "conductor_note": "", "discriminant_digits": 8, "lmfdb": { "label": "1206.b.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C0%2C-2%2C-2%2C4%2C3%5D%2C%5B1%2C1%5D%5D", "snapshot_label": "1206.b.1" }, "sources": [ "Elkies2002" ], "rm": false, "discovered_by": "Elkies 2002", "credit_differs": false, "year": null, "historical": false, "first_source": "Elkies2002", "first_source_year": 2002, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 37 (source: \\cite{Elkies2002})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 17 is strict (χ = T^4 + T^3 + 12*T^2 + 17*T + 289)", "verified": "2026-09-18 17:32:40 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "40.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "66", "0", "12", "0", "2" ], "h": [ "0", "1", "0", "1" ], "equation": { "f": "2*x^4 + 12*x^2 + 66", "h": "x^3 + x", "text": "y^2 + (x^3 + x)*y = 2*x^4 + 12*x^2 + 66" }, "conductor": "3168", "conductor_sort": 3168, "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)", "discriminant_digits": 17, "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" }, "sources": [ "HLP2000" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "HLP2000", "first_source_year": 2000, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 27 (source: \\cite{HLP2000})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:37:24 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "Elkies2002", "qsplit": "HLP2000", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 2, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite", "HLP2000" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4 and §3.5) of E₁ = E₈(t) (torsion ℤ/8), t = (s² − 2s + 3)/(4s² + 4), with E₂ = E₁₀(−1/2) = 66c2 (torsion ℤ/10), along the isomorphism of 2-torsion that matches the rational points of order 2: 8t² − 8t + 1 = −2·□, and (u, y) = (−1/2, (2s² + 2)/(s² − 2s − 1)) is HLP's point on (8t² − 8t + 1)y² = (2u − 1)(4u² − 2u − 1). Up to a square C_s: y² = F(x²), with (x, y) ↦ (x², y) onto E₂ and (x, y) ↦ (1/x², y/x³) onto E₁, so J is isogenous over ℚ to E₁ × E₂. Over ℚ(s) the pullbacks D₈, D₁₀ of the points of order 8 and 10 satisfy 4D₈ = 5D₁₀, and D₈ + 2D₁₀ has order 40. Here #J(𝔽_p) is always divisible by 80, so the group structure is used: every 5- and 31-integral s ≡ 0 (mod 155) has the reductions of s = 0 at 5 and 31, where J(𝔽₅) ≅ ℤ/80 is cyclic and J(𝔽₃₁) ≅ (ℤ/40)², so J(ℚ)_tors is exactly [40]. The member s = 0 is the census curve 40.b; the G2-invariants of the members differ (family_40_split_certificate.log)", "inherited_from": null } } }, { "key": "45", "group": [ 45 ], "bracket": "[45]", "label": "ℤ/45", "order": 45, "rank": 1, "classes": { "simple": [], "qsplit": [ { "id": "45.a", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "168300000000", "0", "49996210000", "0", "29240200", "0", "13981" ], "h": [ "0" ], "equation": { "f": "13981*x^6 + 29240200*x^4 + 49996210000*x^2 + 168300000000", "h": "0", "text": "y^2 = 13981*x^6 + 29240200*x^4 + 49996210000*x^2 + 168300000000" }, "conductor": "602835300", "conductor_sort": 602835300, "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)", "discriminant_digits": 97, "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": "" }, "sources": [ "HLP2000" ], "rm": false, "discovered_by": "Balakrishnan–Najman–Shnidman–Sutherland 2026", "credit_differs": true, "year": null, "historical": false, "first_source": "HLP2000", "first_source_year": 2000, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 28 (source: \\cite{HLP2000})", "new_group": true, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:37:26 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": null, "qsplit": "HLP2000", "gsplit": null, "undecided": null }, "known": { "simple": false, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite", "HLP2000" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4, §3.6) of E = E₉(−5) = 15810u1 (torsion ℤ/9) with the Kubert curves F_t = E₅(t) along a Galois-module isomorphism F_t[2] → E[2], over the curve X′ of pairs (t, ψ), 3-isogenous to HLP's y² = Δ₉(−5)t(t² − 11t − 1): X′ = EM: y² = x³ − 1153200x − 327701000 (conductor 13838400, rank 2), ψ given by a Tschirnhausen transformation with coefficients in ℚ(EM), C_R: y² = c₂F(c₁x² − 1, c₂x²). The maps (x, y) ↦ (1/x², y/x³), (x², y) onto E and F_t make J isogenous over ℚ to E × F_t and give ℤ/45 ⊂ J(ℚ). For R = (1 + 2k)P, P = (1209, −6727) (t = 27/640), the curve has the reduction of C_P at 7, where #J(𝔽₇) = 45, so J(ℚ)_tors is exactly [45]; the G2-invariants at P and 2P differ. The census curve 45.a is the member at (−310, 0) (≡ P mod 7) (family_45_split_certificate.log)", "inherited_from": null } } }, { "key": "48", "group": [ 48 ], "bracket": "[48]", "label": "ℤ/48", "order": 48, "rank": 1, "classes": { "simple": [], "qsplit": [ { "id": "48.a", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "3", "0", "9", "0", "-4" ], "h": [ "0", "1", "0", "1" ], "equation": { "f": "-4*x^4 + 9*x^2 + 3", "h": "x^3 + x", "text": "y^2 + (x^3 + x)*y = -4*x^4 + 9*x^2 + 3" }, "conductor": "1764", "conductor_sort": 1764, "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)", "discriminant_digits": 13, "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" }, "sources": [ "Howe2015", "PP2012b", "Platonov2014" ], "rm": false, "discovered_by": "Platonov–Petrunin 2012b", "credit_differs": true, "year": null, "historical": false, "first_source": "PP2012b", "first_source_year": 2012, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 29 (source: \\cite{Howe2015, PP2012b,Platonov2014})", "new_group": true, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:37:27 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": null, "qsplit": "PP2012b", "gsplit": null, "undecided": null }, "known": { "simple": false, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite", "Howe2015", "HLP2000" ], "note": "Howe 2015, Theorems 3.1 and 3.2: the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4) of the fixed curve F = 42a1: y² = x³ − 31x² + 256x (Q = (32, 96) of order 8) with E_s: y² = x(x² + a_s x + b_s) (P of order 6), Howe's C^s₂₄: d y² = x⁶ + c₄x⁴ + c₂x² + c₀; (x, y) ↦ (x², y) and (x, y) ↦ (1/x², y/x³) map it onto F and E_s, so J is isogenous over ℚ to E_s × F, and the image of (2P, Q) has order 24 over ℚ(s). On Howe's curve D: w² = z³ + 14z² + 196z (9408q1, rank 2), s = −21[(z² + 196)(z² + 56z + 196) + 32(z + 14)(z − 14)w]/(z⁴ − 896z³ − 24696z² − 175616z + 38416), the identity A(B − 32) = ℓ²n² in ℚ(√−7)(D) and HLP Prop. 12 give a rational point of order 16 whose double is the image of (3P, Q), so ℤ/48 ⊂ J(ℚ). For R_k = (1 + 8k)P₂, P₂ = (7, −49), s(R_k) ≡ 3 (mod 5), where J(𝔽₅) = ℤ/48, so J(ℚ)_tors is exactly [48]; the G2-invariants at s = 3 and s = 21/31 differ. The members over P₁, P₂, P₃ are Howe's Corollary 3.5 curves; the census curve 48.a is not a member (its Jacobian has no factor 42a1) (family_48_split_certificate.log)", "inherited_from": null } } }, { "key": "60", "group": [ 60 ], "bracket": "[60]", "label": "ℤ/60", "order": 60, "rank": 1, "classes": { "simple": [], "qsplit": [ { "id": "60.a", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "1", "-7", "25", "-24", "25", "-7", "1" ], "h": [ "0", "1", "1" ], "equation": { "f": "x^6 - 7*x^5 + 25*x^4 - 24*x^3 + 25*x^2 - 7*x + 1", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = x^6 - 7*x^5 + 25*x^4 - 24*x^3 + 25*x^2 - 7*x + 1" }, "conductor": "13500", "conductor_sort": 13500, "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)", "discriminant_digits": 18, "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" }, "sources": [ "HLP2000" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "HLP2000", "first_source_year": 2000, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 30 (source: \\cite{HLP2000})", "new_group": true, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:37:29 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": null, "qsplit": "HLP2000", "gsplit": null, "undecided": null }, "known": { "simple": false, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": { "gsplit": { "reason": "A geometrically split abelian surface that is simple over ℚ has no rational point of order 30. The proof applies to every geometric splitting degree and requires simplicity over ℚ.", "sources": [ "QSimple30Obstruction", "DNQuartic2025", "SerreTate1968", "Tate1966", "Mazur1977", "KenkuMomose1988", "ClarkXarles2008", "Clark2004" ], "proof_url": "proofs/qsimple-30-obstruction.html" } }, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite", "HLP2000" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4, §3.5, equation (3) at t = 1/3) of E = E₁₂(1/3) = 90c3 (torsion ℤ/12) with the Kubert curves F_t = E₁₀(t), over HLP's curve −(5/27)y² = (2t − 1)(4t² − 2t − 1) = EM: y² = x³ − 300x − 1375 (900e1, rank 1), with ψ given by a Tschirnhausen transformation with coefficients in ℚ(EM) and C_R: y² = c₂F(c₁x² − 1, c₂x²). The maps (x, y) ↦ (1/x², y/x³), (x², y) onto E and F_t make J isogenous over ℚ to E × F_t; the pullbacks of the points of order 12 and 10 generate (ℤ/12 × ℤ/10)/⟨(6,5)⟩ = ℤ/60. For R = T + 80kP (T = (−5, 0), t = 2/3, P = (−10, −25)) the curve has the reductions of C_T at 7 and 13, with J(𝔽₇) = ℤ/120 (cyclic 2-part) and J(𝔽₁₃) = [4,60] (2-primary exponent 4), so J(ℚ)_tors is exactly [60]; the G2-invariants at T and 2P differ. The census curve 60.a is the member T (family_60_split_certificate.log)", "inherited_from": null } } }, { "key": "63", "group": [ 63 ], "bracket": "[63]", "label": "ℤ/63", "order": 63, "rank": 1, "classes": { "simple": [], "qsplit": [ { "id": "63.a", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "-146398496", "0", "79136353", "0", "-197570", "0", "897" ], "h": [ "0" ], "equation": { "f": "897*x^6 - 197570*x^4 + 79136353*x^2 - 146398496", "h": "0", "text": "y^2 = 897*x^6 - 197570*x^4 + 79136353*x^2 - 146398496" }, "conductor": "63693396", "conductor_sort": 63693396, "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)", "discriminant_digits": 74, "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": "" }, "sources": [ "HLP2000" ], "rm": false, "discovered_by": "Howe–Leprévost–Poonen 2000", "credit_differs": false, "year": null, "historical": false, "first_source": "HLP2000", "first_source_year": 2000, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 31 (source: \\cite{HLP2000})", "new_group": true, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:37:32 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": null, "qsplit": "HLP2000", "gsplit": null, "undecided": null }, "known": { "simple": false, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": { "gsplit": { "reason": "A geometrically split abelian surface that is simple over ℚ has no rational point of order 63. The proof applies to every geometric splitting degree and requires simplicity over ℚ.", "sources": [ "QSimple63Obstruction", "KenkuMomose1988", "Mazur1977", "DEvHMZB2021", "SerreTate1968", "Clark2004", "ClarkXarles2008" ], "proof_url": "proofs/qsimple-63-obstruction.html" } }, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "open", "sources": [ "HLP2000" ], "note": "HLP 2000, Table 1: a single curve (parameter space ℙ⁰)", "inherited_from": null } } }, { "key": "70", "group": [ 70 ], "bracket": "[70]", "label": "ℤ/70", "order": 70, "rank": 1, "classes": { "simple": [], "qsplit": [ { "id": "70.a", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "179", "425", "-51", "1" ], "h": [ "110", "-41", "-3", "2" ], "equation": { "f": "x^3 - 51*x^2 + 425*x + 179", "h": "2*x^3 - 3*x^2 - 41*x + 110", "text": "y^2 + (2*x^3 - 3*x^2 - 41*x + 110)*y = x^3 - 51*x^2 + 425*x + 179" }, "conductor": "56628", "conductor_sort": 56628, "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", "discriminant_digits": 28, "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": "" }, "sources": [ "Howe2015" ], "rm": false, "discovered_by": "Howe 2015", "credit_differs": false, "year": null, "historical": false, "first_source": "Howe2015", "first_source_year": 2015, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 32 (source: \\cite{Howe2015})", "new_group": true, "new_for_class": true, "certificate": "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]", "verified": "2026-09-18 17:37:33 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": null, "qsplit": "Howe2015", "gsplit": null, "undecided": null }, "known": { "simple": false, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": { "gsplit": { "reason": "A geometrically split abelian surface that is simple over ℚ has no rational point of order 35. The proof applies to every geometric splitting degree and requires simplicity over ℚ.", "sources": [ "QSimple35Obstruction", "DNQuartic2025", "Clark2004", "ClarkXarles2008" ], "proof_url": "proofs/qsimple-35-obstruction.html" } }, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "open", "sources": [], "note": "", "inherited_from": null } } }, { "key": "2.2", "group": [ 2, 2 ], "bracket": "[2,2]", "label": "ℤ/2 ⊕ ℤ/2", "order": 4, "rank": 2, "classes": { "simple": [ { "id": "2.2.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "1", "16", "72", "33", "4" ], "h": [ "0", "1" ], "equation": { "f": "4*x^5 + 33*x^4 + 72*x^3 + 16*x^2 + x", "h": "x", "text": "y^2 + x*y = 4*x^5 + 33*x^4 + 72*x^3 + 16*x^2 + x" }, "conductor": "464", "conductor_sort": 464, "conductor_note": "", "discriminant_digits": 6, "lmfdb": { "label": "464.a.29696.2", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/464/a/29696/2", "snapshot_label": "" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 38 (source: \\cite{BSSVY})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 3 is strict (χ = T^4 + T^3 + 2*T^2 + 3*T + 9)", "verified": "2026-09-18 17:32:41 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "2.2.c", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "0", "3", "28", "72", "28", "3" ], "h": [ "0", "1", "1" ], "equation": { "f": "3*x^5 + 28*x^4 + 72*x^3 + 28*x^2 + 3*x", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = 3*x^5 + 28*x^4 + 72*x^3 + 28*x^2 + 3*x" }, "conductor": "990", "conductor_sort": 990, "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)", "discriminant_digits": 7, "lmfdb": { "label": "990.a.240570.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/990/a/240570/1", "snapshot_label": "" }, "sources": [], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": null, "historical": false, "first_source": null, "first_source_year": null, "torsion_method": "TorsionSubgroup", "reference": "LMFDB 990.a.240570.1, https://www.lmfdb.org/Genus2Curve/Q/990/a/240570/1", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-10-14 10:49:12 UTC", "submitted": "2026-10-14", "source_kind": "issue" } ], "gsplit": [ { "id": "2.2.b", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "-60", "-1", "59", "-1", "59", "0", "-60" ], "h": [ "1", "1", "1", "1" ], "equation": { "f": "-60*x^6 + 59*x^4 - x^3 + 59*x^2 - x - 60", "h": "x^3 + x^2 + x + 1", "text": "y^2 + (x^3 + x^2 + x + 1)*y = -60*x^6 + 59*x^4 - x^3 + 59*x^2 - x - 60" }, "conductor": "256", "conductor_sort": 256, "conductor_note": "", "discriminant_digits": 20, "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" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 33 (source: \\cite{BSSVY})", "new_group": false, "new_for_class": true, "certificate": "the geometric automorphism group has order 8 (5 involutions), so the curve is bielliptic over Qbar; simple over ℚ: χ_3 = T^4 + 2*T^3 + 2*T^2 + 6*T + 9 is irreducible", "verified": "2026-09-18 17:37:34 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "BSSVY", "qsplit": null, "gsplit": "BSSVY", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite" ], "note": "y² = x(x−1)(x−2)(x³ + x + t): [x(x−1), 0] and [x(x−2), 0] mark [2,2] over ℚ(t). Every t ≡ 3 (mod 119) has #J(𝔽₇) = 72 and #J(𝔽₁₇) = 316 (gcd 4), strict at 17 (family_2_2_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings, in the normal form of Bröker–Howe–Lauter–Stevenhagen (arXiv:1403.6911), of E = 17a2 (torsion ℤ/2 × ℤ/2) with all partners F anti-symplectically 3-congruent to E. These partners form a rational curve (the twist X_E^−(3) of X(3)); its parameter w gives C_w over ℚ(w) with explicit degree-3 maps onto E and F_w, so J is isogenous over ℚ to E × F_w. The map onto E has prime degree, so pullback is injective on E and [2,2] ⊂ J(ℚ). Every 7- and 13-integral w ≡ 2 (mod 91) has the reductions of w = 2 at 7 and 13, where J(𝔽₇) ≅ [2,2,10] and J(𝔽₁₃) ≅ [4,76]; the largest group embedding in both is [2,2], so J(ℚ)_tors is exactly [2,2]; the G2-invariants at w = 2 and w = −89 differ (family_2_2_split_certificate.log)", "inherited_from": null } } }, { "key": "2.4", "group": [ 2, 4 ], "bracket": "[2,4]", "label": "ℤ/2 ⊕ ℤ/4", "order": 8, "rank": 2, "classes": { "simple": [ { "id": "2.4.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "-1", "0", "16", "-8", "1" ], "h": [ "0", "1" ], "equation": { "f": "x^5 - 8*x^4 + 16*x^3 - x", "h": "x", "text": "y^2 + x*y = x^5 - 8*x^4 + 16*x^3 - x" }, "conductor": "997", "conductor_sort": 997, "conductor_note": "", "discriminant_digits": 6, "lmfdb": { "label": "997.a.997.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/997/a/997/1", "snapshot_label": "" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 39 (source: \\cite{BSSVY})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 3 is strict (χ = T^4 + T^3 + 2*T^2 + 3*T + 9)", "verified": "2026-09-18 17:32:42 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "2.4.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "-16", "6", "-6", "11", "1", "3" ], "h": [ "1", "0", "1" ], "equation": { "f": "3*x^5 + x^4 + 11*x^3 - 6*x^2 + 6*x - 16", "h": "x^2 + 1", "text": "y^2 + (x^2 + 1)*y = 3*x^5 + x^4 + 11*x^3 - 6*x^2 + 6*x - 16" }, "conductor": "225", "conductor_sort": 225, "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", "discriminant_digits": 10, "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" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 34 (source: \\cite{BSSVY})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:37:36 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [ { "id": "2.4.c", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "0", "-1", "11", "35", "-123", "-171", "405" ], "h": [ "0" ], "equation": { "f": "405*x^6 - 171*x^5 - 123*x^4 + 35*x^3 + 11*x^2 - x", "h": "0", "text": "y^2 = 405*x^6 - 171*x^5 - 123*x^4 + 35*x^3 + 11*x^2 - x" }, "conductor": "2457", "conductor_sort": 2457, "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)", "discriminant_digits": 19, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": 2026, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "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)", "new_group": false, "new_for_class": true, "certificate": "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); simple over ℚ: χ_11 = T^4 - 10*T^2 + 121 is irreducible", "verified": "2026-10-17 06:12:08 UTC", "submitted": "2026-10-16", "source_kind": "file" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": 0, "undecided": null }, "first_source": { "simple": "BSSVY", "qsplit": "BSSVY", "gsplit": "BookerSutherland", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite" ], "note": "y² = x(x² + (2t+1)x + t²)(x² + 3x + 1): D = [x² + tx + t, x(2x + t + 1) mod u] has 2D = [x, 0] ≠ 0, and with T = [x² + 3x + 1, 0] it marks [2,4] over ℚ(t). Every t ≡ 3 (mod 119) has #J(𝔽₇) = 56 and #J(𝔽₁₇) = 368 (gcd 8), strict at 17 (family_2_4_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings, in the normal form of Bröker–Howe–Lauter–Stevenhagen (arXiv:1403.6911), of E = 15a1 (torsion ℤ/2 × ℤ/4) with all partners F anti-symplectically 3-congruent to E. These partners form a rational curve (the twist X_E^−(3) of X(3)); its parameter w gives C_w over ℚ(w) with explicit degree-3 maps onto E and F_w, so J is isogenous over ℚ to E × F_w. The map onto E has prime degree, so pullback is injective on E and [2,4] ⊂ J(ℚ). Every 11- and 13-integral w ≡ −3 (mod 143) has the reductions of w = −3 at 11 and 13, where J(𝔽₁₁) ≅ [2,104] and J(𝔽₁₃) ≅ [4,76]; the largest group embedding in both is [2,4], so J(ℚ)_tors is exactly [2,4]; the G2-invariants at w = −3 and w = 140 differ (family_2_4_split_certificate.log)", "inherited_from": null } } }, { "key": "2.6", "group": [ 2, 6 ], "bracket": "[2,6]", "label": "ℤ/2 ⊕ ℤ/6", "order": 12, "rank": 2, "classes": { "simple": [ { "id": "2.6.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "0", "-2", "-1", "4", "4" ], "h": [ "1" ], "equation": { "f": "4*x^5 + 4*x^4 - x^3 - 2*x^2", "h": "1", "text": "y^2 + y = 4*x^5 + 4*x^4 - x^3 - 2*x^2" }, "conductor": "704", "conductor_sort": 704, "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", "discriminant_digits": 6, "lmfdb": { "label": "704.a.45056.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/704/a/45056/1", "snapshot_label": "" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 40 (source: \\cite{BSSVY})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 23 is strict (χ = T^4 - 3*T^3 - 2*T^2 - 69*T + 529)", "verified": "2026-09-18 17:32:42 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "2.6.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "-23", "13", "-14", "-3", "1", "-2" ], "h": [ "0", "1", "1" ], "equation": { "f": "-2*x^5 + x^4 - 3*x^3 - 14*x^2 + 13*x - 23", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = -2*x^5 + x^4 - 3*x^3 - 14*x^2 + 13*x - 23" }, "conductor": "196", "conductor_sort": 196, "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)", "discriminant_digits": 13, "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" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 35 (source: \\cite{BSSVY})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:37:37 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [ { "id": "2.6.c", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "0", "1", "-4", "-13", "-9", "-1" ], "h": [ "0" ], "equation": { "f": "-x^5 - 9*x^4 - 13*x^3 - 4*x^2 + x", "h": "0", "text": "y^2 = -x^5 - 9*x^4 - 13*x^3 - 4*x^2 + x" }, "conductor": "784", "conductor_sort": 784, "conductor_note": "", "discriminant_digits": 9, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": 2026, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "generators", "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)", "new_group": false, "new_for_class": true, "certificate": "the geometric automorphism group has order 12 (7 involutions), so the curve is bielliptic over Qbar; simple over ℚ: χ_3 = T^4 + T^3 - 2*T^2 + 3*T + 9 is irreducible", "verified": "2026-10-17 06:12:10 UTC", "submitted": "2026-10-16", "source_kind": "file" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": 0, "undecided": null }, "first_source": { "simple": "BSSVY", "qsplit": "BSSVY", "gsplit": "BookerSutherland", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite" ], "note": "contact-6 chart f = h² − (x−1)⁶ with h = 1 + tx + x² + x³ (h(0) = 1 gives the root x = 0): D₆ = [(1, h(1)) − ∞] has order 6 and [x, 0] is a class of order 2 outside ⟨D₆⟩, so [2,6] is marked over ℚ(t). Every t ≡ 13 (mod 39) has #J(𝔽₃) = 12 and #J(𝔽₁₃) = 120 (gcd 12), strict at 13 (family_2_6_certificate.log). The record's 'h cubic' needs h(0) = 1 for the class of order 2", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings, in the normal form of Bröker–Howe–Lauter–Stevenhagen (arXiv:1403.6911), of E = 30a2 (torsion ℤ/2 × ℤ/6) with all partners F anti-symplectically 3-congruent to E. These partners form a rational curve (the twist X_E^−(3) of X(3)); its parameter w gives C_w over ℚ(w) with explicit degree-3 maps onto E and F_w, so J is isogenous over ℚ to E × F_w. The map onto E has prime degree, so pullback is injective on E and [2,6] ⊂ J(ℚ); F_w also inherits a rational point of order 3 from E, but no extra 3-torsion appears in J on the class below. Every 11- and 13-integral w ≡ −4 (mod 143) has the reductions of w = −4 at 11 and 13, where J(𝔽₁₁) ≅ [2,54] and J(𝔽₁₃) ≅ [2,6,12]; the largest group embedding in both is [2,6], so J(ℚ)_tors is exactly [2,6]; the G2-invariants at w = −4 and w = 139 differ (family_2_6_split_certificate.log)", "inherited_from": null } } }, { "key": "2.8", "group": [ 2, 8 ], "bracket": "[2,8]", "label": "ℤ/2 ⊕ ℤ/8", "order": 16, "rank": 2, "classes": { "simple": [ { "id": "2.8.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "0", "-2", "-4", "3", "8" ], "h": [ "1", "1" ], "equation": { "f": "8*x^5 + 3*x^4 - 4*x^3 - 2*x^2", "h": "x + 1", "text": "y^2 + (x + 1)*y = 8*x^5 + 3*x^4 - 4*x^3 - 2*x^2" }, "conductor": "464", "conductor_sort": 464, "conductor_note": "", "discriminant_digits": 6, "lmfdb": { "label": "464.a.29696.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/464/a/29696/1", "snapshot_label": "" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 41 (source: \\cite{BSSVY})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 3 is strict (χ = T^4 + T^3 + 2*T^2 + 3*T + 9)", "verified": "2026-09-18 17:32:43 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "2.8.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "-214", "465", "-279", "447", "-305", "60", "-175" ], "h": [ "1", "0", "1" ], "equation": { "f": "-175*x^6 + 60*x^5 - 305*x^4 + 447*x^3 - 279*x^2 + 465*x - 214", "h": "x^2 + 1", "text": "y^2 + (x^2 + 1)*y = -175*x^6 + 60*x^5 - 305*x^4 + 447*x^3 - 279*x^2 + 465*x - 214" }, "conductor": "225", "conductor_sort": 225, "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", "discriminant_digits": 34, "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" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 36 (source: \\cite{BSSVY})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:37:39 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [ { "id": "2.8.c", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "-8152375", "2562175", "73961", "83792", "-1345", "847", "49" ], "h": [ "0", "1", "1" ], "equation": { "f": "49*x^6 + 847*x^5 - 1345*x^4 + 83792*x^3 + 73961*x^2 + 2562175*x - 8152375", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = 49*x^6 + 847*x^5 - 1345*x^4 + 83792*x^3 + 73961*x^2 + 2562175*x - 8152375" }, "conductor": "979254391500", "conductor_sort": 979254391500, "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", "discriminant_digits": 57, "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": "" }, "sources": [], "rm": false, "discovered_by": "F. Najman", "credit_differs": false, "year": 2026, "historical": false, "first_source": null, "first_source_year": null, "torsion_method": "TorsionSubgroup", "reference": "new", "new_group": false, "new_for_class": true, "certificate": "(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); simple over ℚ: χ_19 = T^4 - 10*T^2 + 361 is irreducible", "verified": "2026-10-17 06:12:13 UTC", "submitted": "2026-10-16", "source_kind": "file" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": 0, "undecided": null }, "first_source": { "simple": "BSSVY", "qsplit": "BSSVY", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite" ], "note": "the Richelot quotient of the [2,2,2,8] family by the kernel K = {[(−b²,0) − ∞], [(0,0) − (−a²,0)], [(−1,0) − (−t²,0)]}, the record's 'partner #4' at t = 2: the pushforward of the class of order 8 and a class of order 2 mark [2,8] over ℚ(t). Every t ≡ 2 (mod 82739) has J′(𝔽₁₇) ≅ [16,16], J′(𝔽₃₁) ≅ [2,2,272] and J′(𝔽₁₅₇) ≅ [4,24,264]; the only abelian 2-group containing [2,8] that embeds in all three is [2,8], and the odd parts have gcd 1, so J′(ℚ)_tors is exactly [2,8]; strict at 31 (family_2_8_certificate.log). The record's 'partner #4' is not one family: Magma orders the Richelot partners by the size of the roots, and partner #4 is K at t = 2, 3 but {[(−a²,0) − ∞], [(0,0) − (−b²,0)], [(−1,0) − (−t²,0)]} at t = 5, 7 (audit of 2026-10-18)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings, in the normal form of Bröker–Howe–Lauter–Stevenhagen (arXiv:1403.6911), of E = 210e2 (torsion ℤ/2 × ℤ/8) with all partners F anti-symplectically 3-congruent to E. These partners form a rational curve (the twist X_E^−(3) of X(3)); its parameter w gives C_w over ℚ(w) with explicit degree-3 maps onto E and F_w, so J is isogenous over ℚ to E × F_w. The map onto E has prime degree, so pullback is injective on E and [2,8] ⊂ J(ℚ). Every 11- and 13-integral w ≡ −1 (mod 143) has the reductions of w = −1 at 11 and 13, where J(𝔽₁₁) ≅ [2,56] and J(𝔽₁₃) ≅ [2,8,16]; the largest group embedding in both is [2,8], so J(ℚ)_tors is exactly [2,8]; the G2-invariants at w = −1 and w = 142 differ (family_2_8_split_certificate.log)", "inherited_from": null } } }, { "key": "2.10", "group": [ 2, 10 ], "bracket": "[2,10]", "label": "ℤ/2 ⊕ ℤ/10", "order": 20, "rank": 2, "classes": { "simple": [ { "id": "2.10.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "1", "1", "-4", "-2", "3" ], "h": [ "1", "1" ], "equation": { "f": "3*x^5 - 2*x^4 - 4*x^3 + x^2 + x", "h": "x + 1", "text": "y^2 + (x + 1)*y = 3*x^5 - 2*x^4 - 4*x^3 + x^2 + x" }, "conductor": "555", "conductor_sort": 555, "conductor_note": "", "discriminant_digits": 6, "lmfdb": { "label": "555.a.8325.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/555/a/8325/1", "snapshot_label": "" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 42 (source: \\cite{BSSVY})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 7 is strict (χ = T^4 - T^3 - 2*T^2 - 7*T + 49)", "verified": "2026-09-18 17:32:43 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "2.10.c", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "0", "2", "4", "2", "0", "1" ], "h": [ "1", "0", "1" ], "equation": { "f": "x^5 + 2*x^3 + 4*x^2 + 2*x", "h": "x^2 + 1", "text": "y^2 + (x^2 + 1)*y = x^5 + 2*x^3 + 4*x^2 + 2*x" }, "conductor": "363", "conductor_sort": 363, "conductor_note": "", "discriminant_digits": 7, "lmfdb": { "label": "363.a.11979.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/363/a/11979/1", "snapshot_label": "" }, "sources": [], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": null, "historical": false, "first_source": null, "first_source_year": null, "torsion_method": "TorsionSubgroup", "reference": "LMFDB 363.a.11979.1, https://www.lmfdb.org/Genus2Curve/Q/363/a/11979/1", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-10-14 10:49:17 UTC", "submitted": "2026-10-14", "source_kind": "issue" } ], "gsplit": [ { "id": "2.10.b", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "0", "0", "-1", "-1", "-1", "-1" ], "h": [ "1", "1", "1", "1" ], "equation": { "f": "-x^5 - x^4 - x^3 - x^2", "h": "x^3 + x^2 + x + 1", "text": "y^2 + (x^3 + x^2 + x + 1)*y = -x^5 - x^4 - x^3 - x^2" }, "conductor": "256", "conductor_sort": 256, "conductor_note": "", "discriminant_digits": 7, "lmfdb": { "label": "256.a.512.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/256/a/512/1", "snapshot_label": "" }, "sources": [ "Ogg1973" ], "rm": false, "discovered_by": "Ogg 1973", "credit_differs": false, "year": null, "historical": false, "first_source": "Ogg1973", "first_source_year": 1973, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 37 (source: \\cite{Ogg1973})", "new_group": false, "new_for_class": true, "certificate": "the geometric automorphism group has order 8 (5 involutions), so the curve is bielliptic over Qbar; simple over ℚ: χ_3 = T^4 + 2*T^3 + 2*T^2 + 6*T + 9 is irreducible", "verified": "2026-09-18 17:37:40 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "BSSVY", "qsplit": null, "gsplit": "Ogg1973", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite", "Elkies2024" ], "note": "Elkies' 5-torsion chart with three rational Weierstrass points, slice (t, 2, 3): D₅ = [Q, x] has order 5 and two root-pair classes generate (ℤ/2)², so [2,10] is marked over ℚ(t). Every t ≡ 31 (mod 85) has #J(𝔽₅) = 40 and #J(𝔽₁₇) = 260 (gcd 20), strict at 17 (family_210_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4) of the Kubert curve E₂ = E₁₀(t) (torsion ℤ/10) with its 5-isogenous curve E₁ = E₂/⟨2P⟩, along the isomorphism of 2-torsion induced by the isogeny. Up to a square it is C_t: y² = F(x²), with (x, y) ↦ (x², y) onto E₂ and (x, y) ↦ (1/x², y/x³) onto E₁, so J is isogenous over ℚ to E₁ × E₂. Over ℚ(t) two distinct classes of order 2 (one of them the gain of the gluing: the Kummer maps commute with the isogeny) and the pullback of 2P, of order 5, mark [2,10]. Every 7- and 11-integral t ≡ 58 (mod 77) has J(𝔽₇) ≅ J(𝔽₁₁) ≅ [10,10]; the kernel of E₁ → E₂ is μ₅ and 11 ≡ 1 (mod 5), so a rational point of order 5 on E₁ would give J(𝔽₁₁)[5] rank ≥ 3, and J(ℚ)_tors is exactly [2,10]; the G2-invariants of the members differ (family_210_split_certificate.log)", "inherited_from": null } } }, { "key": "2.12", "group": [ 2, 12 ], "bracket": "[2,12]", "label": "ℤ/2 ⊕ ℤ/12", "order": 24, "rank": 2, "classes": { "simple": [ { "id": "2.12.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "-1", "2", "14", "-8", "1" ], "h": [ "0", "1", "1" ], "equation": { "f": "x^5 - 8*x^4 + 14*x^3 + 2*x^2 - x", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = x^5 - 8*x^4 + 14*x^3 + 2*x^2 - x" }, "conductor": "762", "conductor_sort": 762, "conductor_note": "", "discriminant_digits": 8, "lmfdb": { "label": "762.a.82296.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/762/a/82296/1", "snapshot_label": "" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 43 (source: \\cite{BSSVY})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 7 is strict (χ = T^4 - T^3 + 6*T^2 - 7*T + 49)", "verified": "2026-09-18 17:32:44 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "2.12.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "-5", "15", "-8", "-10", "4", "2" ], "h": [ "0", "1", "1", "1" ], "equation": { "f": "2*x^5 + 4*x^4 - 10*x^3 - 8*x^2 + 15*x - 5", "h": "x^3 + x^2 + x", "text": "y^2 + (x^3 + x^2 + x)*y = 2*x^5 + 4*x^4 - 10*x^3 - 8*x^2 + 15*x - 5" }, "conductor": "450", "conductor_sort": 450, "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)", "discriminant_digits": 9, "lmfdb": { "label": "450.a.36450.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/450/a/36450/1", "snapshot_label": "" }, "sources": [ "HLP2000" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": true, "year": null, "historical": false, "first_source": "HLP2000", "first_source_year": 2000, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 38 (source: \\cite{HLP2000})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:37:41 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "BSSVY", "qsplit": "HLP2000", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 2, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite" ], "note": "the line a = (1−r)/4 in the M(12) chart: [P − (2, 0)] has order 12, and the root u = −2r/(r−2) gives a class [(x−u)(x−2), 0] of order 2 outside it, so [2,12] is marked over ℚ(r). Every r ≡ 2 (mod 35), r ≠ 2, has #J(𝔽₅) = 24 and #J(𝔽₇) = 48 (gcd 24), strict at 7 (family_2_12_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings (Bröker–Howe–Lauter–Stevenhagen normal form) of E = 90c3 (torsion ℤ/12) with the partners F that have a rational 2-torsion point. The partners anti-symplectically 3-congruent to E form a rational curve, and so does the cover on which F has a rational 2-torsion point (Γ(3) ∩ Γ₀(2) has genus 0); this gives C_v over ℚ(v) with explicit degree-3 maps onto E and F_v, so J is isogenous over ℚ to E × F_v. The map onto E has prime degree, so pullback is injective on E and [12] ⊂ J(ℚ); over ℚ(v) the quadratic divisors of f_v give J(ℚ)[2] of 2-rank 2, one more, so [2,12] is marked. Every 7- and 11-integral v ≡ 4 (mod 77) has the reductions of v = 4 at 7 and 11, where J(𝔽₇) ≅ [2,36] and J(𝔽₁₁) ≅ [12,12]; the largest group embedding in both is [2,12], so J(ℚ)_tors is exactly [2,12]; the G2-invariants at v = 4 and v = −73 differ (family_2_12_split_certificate.log)", "inherited_from": null } } }, { "key": "2.14", "group": [ 2, 14 ], "bracket": "[2,14]", "label": "ℤ/2 ⊕ ℤ/14", "order": 28, "rank": 2, "classes": { "simple": [ { "id": "2.14.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "1", "1", "0", "-1", "-2" ], "h": [ "0", "1", "0", "1" ], "equation": { "f": "-2*x^4 - x^3 + x + 1", "h": "x^3 + x", "text": "y^2 + (x^3 + x)*y = -2*x^4 - x^3 + x + 1" }, "conductor": "1416", "conductor_sort": 1416, "conductor_note": "", "discriminant_digits": 9, "lmfdb": { "label": "1416.b.135936.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/1416/b/135936/1", "snapshot_label": "" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 44 (source: \\cite{BSSVY})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 17 is strict (χ = T^4 - 4*T^3 + 6*T^2 - 68*T + 289)", "verified": "2026-09-18 17:32:44 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "2.14.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "10", "-6", "-24", "-5", "-9", "-3", "9" ], "h": [ "0", "1", "1" ], "equation": { "f": "9*x^6 - 3*x^5 - 9*x^4 - 5*x^3 - 24*x^2 - 6*x + 10", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = 9*x^6 - 3*x^5 - 9*x^4 - 5*x^3 - 24*x^2 - 6*x + 10" }, "conductor": "86436", "conductor_sort": 86436, "conductor_note": "", "discriminant_digits": 20, "lmfdb": { "label": "86436.z.3", "kind": "alpha-snapshot", "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 39 (source: \\cite{BookerSutherland})", "new_group": false, "new_for_class": true, "certificate": "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]", "verified": "2026-09-18 17:37:43 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "BSSVY", "qsplit": "BookerSutherland", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 2, "infinite": { "simple": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the contact-7 two-root surface (h = 1 − (7/2)x + ax² + bx³, f = (h² + (x−1)⁷)/x², with roots 1 − s², 1 − t²), slice s = 1: [2,14] is marked over ℚ(t). Every t ≡ 2 (mod 65) has #J(𝔽₅) = 28, and a strict Frobenius at 13 (family_214_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings (Bröker–Howe–Lauter–Stevenhagen normal form) of E = 294b2 (torsion ℤ/7) with the partners F that have full rational 2-torsion. They are parametrised by the genus-1 curve D: s² = 6769v⁴ − 2324v³ − 714v² − 644v + 49, whose Jacobian y² = x³ + 86436 has the point P = (−392/9, 1666/27) of infinite order. Explicit degree-3 maps onto E and F_v make J isogenous over ℚ to E × F_v, and pullback is injective on torsion of order prime to 3, so [2,14] ⊂ J(ℚ). For R_k = R₀ + 124kP (R₀ over v = 0, the census curve 2.14.b, the gluing of 294b2 and 294c4), v(R_k) ≡ 0 modulo 11 and 97, where #J(𝔽₁₁) = 112 and #J(𝔽₉₇) = 8820, so J(ℚ)_tors is exactly [2,14]; the G2-invariants at v = 0 and v = 34/825 differ (family_214_split_certificate.log)", "inherited_from": null } } }, { "key": "2.16", "group": [ 2, 16 ], "bracket": "[2,16]", "label": "ℤ/2 ⊕ ℤ/16", "order": 32, "rank": 2, "classes": { "simple": [ { "id": "2.16.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "8", "4", "-3", "1", "-1", "-1" ], "h": [ "0", "0", "1", "1" ], "equation": { "f": "-x^5 - x^4 + x^3 - 3*x^2 + 4*x + 8", "h": "x^3 + x^2", "text": "y^2 + (x^3 + x^2)*y = -x^5 - x^4 + x^3 - 3*x^2 + 4*x + 8" }, "conductor": "4392", "conductor_sort": 4392, "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", "discriminant_digits": 12, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 45 (source: \\cite{BookerSutherland})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 5 is strict (χ = T^4 + T^3 + 5*T + 25)", "verified": "2026-09-18 17:32:45 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "2.16.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "2", "7", "16", "18", "16", "7", "2" ], "h": [ "1", "1", "1", "1" ], "equation": { "f": "2*x^6 + 7*x^5 + 16*x^4 + 18*x^3 + 16*x^2 + 7*x + 2", "h": "x^3 + x^2 + x + 1", "text": "y^2 + (x^3 + x^2 + x + 1)*y = 2*x^6 + 7*x^5 + 16*x^4 + 18*x^3 + 16*x^2 + 7*x + 2" }, "conductor": "4608", "conductor_sort": 4608, "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)", "discriminant_digits": 13, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 40 (source: \\cite{BookerSutherland})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:37:45 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "BookerSutherland", "qsplit": "BookerSutherland", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 2, "infinite": { "simple": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the [16] family at s = 3/4 + u², where f has two rational roots: [2,16] is marked over ℚ(u). Every 5-integral u ≡ 1 (mod 5) has #J(𝔽₅) = 32, strict (family_216_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4) of E₁ = 210e2 (torsion ℤ/2 × ℤ/8, P = (244, 3658), monic roots −36, 31/4, 28) with the curves E₂(s): Y² = X(X + 70s²)(X + 105), along (−36, 31/4, 28) ↦ (−70s², −105, 0): the Kummer class of (0, 0) on E₂(s) is (70, 105, 6), that of P, which is the gain of the gluing. Up to a square C_s: y² = (128 − 27s²)((8s² − 12)x² − 5)(35s²x² − 32)(140x² − 27), with (x, y) ↦ (x², y) onto E₂(s) and (x, y) ↦ (1/x², y/x³) onto E₁, so J is isogenous over ℚ to E₁ × E₂(s). The sum of the pullbacks of halves of P and of (0, 0), computed over ℚ(√70, √105)(s), has coefficients in ℚ(s) and order 16; with the pullback of (−36, 18) it marks [2,16] over ℚ(s). Here #J(𝔽_p) is always divisible by 64, so the group structure is used instead: every 11- and 13-integral s ≡ 67 (mod 143) has J(𝔽₁₁) ≅ [8,16] and J(𝔽₁₃) ≅ [2,2,80], so J(ℚ)_tors is exactly [2,16]; the G2-invariants of the members differ (family_2_16_split_certificate.log)", "inherited_from": null } } }, { "key": "2.18", "group": [ 2, 18 ], "bracket": "[2,18]", "label": "ℤ/2 ⊕ ℤ/18", "order": 36, "rank": 2, "classes": { "simple": [ { "id": "2.18.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "7", "2", "1", "3", "-5", "-1", "1" ], "h": [ "0", "1", "1" ], "equation": { "f": "x^6 - x^5 - 5*x^4 + 3*x^3 + x^2 + 2*x + 7", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = x^6 - x^5 - 5*x^4 + 3*x^3 + x^2 + 2*x + 7" }, "conductor": "4428", "conductor_sort": 4428, "conductor_note": "", "discriminant_digits": 14, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 46 (source: \\cite{BookerSutherland})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 17 is strict (χ = T^4 + T^3 + 16*T^2 + 17*T + 289)", "verified": "2026-09-18 17:32:46 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "2.18.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "36", "66", "12", "-5", "-3", "-3", "1" ], "h": [ "0", "1", "1" ], "equation": { "f": "x^6 - 3*x^5 - 3*x^4 - 5*x^3 + 12*x^2 + 66*x + 36", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = x^6 - 3*x^5 - 3*x^4 - 5*x^3 + 12*x^2 + 66*x + 36" }, "conductor": "64980", "conductor_sort": 64980, "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", "discriminant_digits": 18, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 41 (source: \\cite{BookerSutherland})", "new_group": false, "new_for_class": true, "certificate": "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]", "verified": "2026-09-18 17:37:46 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "BookerSutherland", "qsplit": "BookerSutherland", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 2, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings (Bröker–Howe–Lauter–Stevenhagen normal form) of E = 714i1 (torsion ℤ/9) with the partners F that have full rational 2-torsion. They are parametrised by the genus-1 curve D: s² = −5183v⁴ − 60435076v³ − 234870845178v² − 304157704212868v + 304392635498305, whose Jacobian y² = x³ + 289 has the point P = (−4, −15) of infinite order (for 54b3 this Jacobian is y² = x³ + 81, of rank 0). Explicit degree-3 maps make J isogenous over ℚ to E × F_v; pullback along the map onto E is injective (prime degree) and along the map onto F_v is injective on F_v(ℚ)[2], so [2,18] ⊂ J(ℚ). For R_k = R₀ + 18kP (R₀ = P + (0, 17) over v = −3023), v(R_k) ≡ −3023 modulo 109, 251 and 359, where J(𝔽₁₀₉) ≅ [2,2,54,54] (2-primary exponent 2), J(𝔽₂₅₁) ≅ [2,31320] (3-primary part ℤ/27) and J(𝔽₃₅₉) ≅ [2,2,180,180] (3-primary exponent 9), so J(ℚ)_tors is exactly [2,18]; the G2-invariants at v = −3023 and at v = −11563693/3107 (over 2P) differ (family_218_split_certificate.log)", "inherited_from": null } } }, { "key": "2.20", "group": [ 2, 20 ], "bracket": "[2,20]", "label": "ℤ/2 ⊕ ℤ/20", "order": 40, "rank": 2, "classes": { "simple": [ { "id": "2.20.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "-180", "-251", "185", "28", "-30", "4" ], "h": [ "0", "0", "1" ], "equation": { "f": "4*x^6 - 30*x^5 + 28*x^4 + 185*x^3 - 251*x^2 - 180*x", "h": "x^2", "text": "y^2 + x^2*y = 4*x^6 - 30*x^5 + 28*x^4 + 185*x^3 - 251*x^2 - 180*x" }, "conductor": "936600", "conductor_sort": 936600, "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", "discriminant_digits": 23, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 47 (source: \\cite{BookerSutherland})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 17 is strict (χ = T^4 - 3*T^3 + 4*T^2 - 51*T + 289)", "verified": "2026-09-18 17:32:46 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "2.20.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "0", "-18", "42", "-20", "15", "-3", "1" ], "h": [ "0", "1", "1" ], "equation": { "f": "x^6 - 3*x^5 + 15*x^4 - 20*x^3 + 42*x^2 - 18*x", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = x^6 - 3*x^5 + 15*x^4 - 20*x^3 + 42*x^2 - 18*x" }, "conductor": "30492", "conductor_sort": 30492, "conductor_note": "", "discriminant_digits": 18, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 42 (source: \\cite{BookerSutherland})", "new_group": false, "new_for_class": true, "certificate": "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]", "verified": "2026-09-18 17:37:48 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "BookerSutherland", "qsplit": "BookerSutherland", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 2, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite" ], "note": "the extra-2 locus t = −(z⁴ + 4z + 4)/(z⁴ + 4z³ + 8z² + 8z + 4) of the [20] family, where f_t has the second rational root z/(t+1): [x − z/(t+1), 0] is a class of order 2 outside ⟨D₅, H⟩, so [2,20] is marked over ℚ(z). Every 7-integral z ≡ 1 (mod 7) has #J(𝔽₇) = 40, strict (family_220_certificate.log)", "inherited_from": null }, "split": { "grade": "open", "sources": [], "note": "", "inherited_from": null } } }, { "key": "2.22", "group": [ 2, 22 ], "bracket": "[2,22]", "label": "ℤ/2 ⊕ ℤ/22", "order": 44, "rank": 2, "classes": { "simple": [ { "id": "2.22.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "-6", "12", "-5", "9", "-3", "1" ], "h": [ "0", "1", "1" ], "equation": { "f": "x^6 - 3*x^5 + 9*x^4 - 5*x^3 + 12*x^2 - 6*x", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = x^6 - 3*x^5 + 9*x^4 - 5*x^3 + 12*x^2 - 6*x" }, "conductor": "19044", "conductor_sort": 19044, "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", "discriminant_digits": 16, "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" }, "sources": [ "BookerSutherland" ], "rm": true, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 48 (source: \\cite{BookerSutherland}$^{\\rm RM}$)", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 5 is strict (χ = T^4 + 2*T^3 + 6*T^2 + 10*T + 25)", "verified": "2026-09-18 17:32:47 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "BookerSutherland", "qsplit": null, "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": false, "gsplit": false, "undecided": false }, "exclusions": { "qsplit": { "reason": "A Jacobian split over ℚ has no rational torsion of prime order greater than 7, by Mazur's theorem applied to an elliptic subvariety and its elliptic quotient.", "sources": [ "QSplitPrimeObstruction", "Mazur1977" ], "proof_url": "proofs/qsplit-prime-obstruction.html" }, "gsplit": { "reason": "A geometrically split abelian surface over ℚ has no rational point of order 22. The proof applies to every geometric splitting degree.", "sources": [ "Geometric22Obstruction", "Mazur1977", "KenkuMomose1988", "LeFournNajman2020", "BourdonClark2020" ], "proof_url": "proofs/geometric-22-obstruction.html" } }, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [ "BNSS2026", "Najman2026", "Sutherland2026pc" ], "note": "two curves with real multiplication by ℤ[φ] (φ² = φ + 1) are known, and one without: A. Sutherland's curve y² + (x² + x)y = 9x⁶ + 111x⁵ + 279x⁴ − 485x³ − 282x² + 1836x + 1456 (conductor 30066300) has End(J_ℚ̄) = ℤ; its torsion and End(J_ℚ̄) = ℤ (the Frobenius fields at 13 and 19 are quartic CM fields with no common quadratic subfield) are checked in sutherland_2_22_check.log", "inherited_from": null }, "split": { "grade": "impossible", "sources": [ "Geometric22Obstruction", "Mazur1977", "KenkuMomose1988", "LeFournNajman2020", "BourdonClark2020" ], "note": "A geometrically split abelian surface over ℚ has no rational point of order 22. The proof applies to every geometric splitting degree.", "proof_url": "proofs/geometric-22-obstruction.html", "inherited_from": null } } }, { "key": "2.24", "group": [ 2, 24 ], "bracket": "[2,24]", "label": "ℤ/2 ⊕ ℤ/24", "order": 48, "rank": 2, "classes": { "simple": [ { "id": "2.24.b", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "11816100", "-18141270", "-1242978", "27097", "6942", "588", "16" ], "h": [ "0", "1", "1" ], "equation": { "f": "16*x^6 + 588*x^5 + 6942*x^4 + 27097*x^3 - 1242978*x^2 - 18141270*x + 11816100", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = 16*x^6 + 588*x^5 + 6942*x^4 + 27097*x^3 - 1242978*x^2 - 18141270*x + 11816100" }, "conductor": "116250940866060", "conductor_sort": 116250940866060, "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)", "discriminant_digits": 58, "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": "" }, "sources": [], "rm": false, "discovered_by": "A. V. Sutherland", "credit_differs": false, "year": 2026, "historical": false, "first_source": null, "first_source_year": null, "torsion_method": "TorsionSubgroup", "reference": "A. V. Sutherland, personal communication to F. Najman (November 2026)", "new_group": false, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 29 is strict (χ = T^4 - 2*T^3 - 14*T^2 - 58*T + 841)", "verified": "2026-11-10 12:02:24 UTC", "submitted": "2026-11-10", "source_kind": "file" } ], "qsplit": [ { "id": "2.24.a", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "15", "0", "-6", "0", "-1" ], "h": [ "0", "1", "0", "1" ], "equation": { "f": "-x^4 - 6*x^2 + 15", "h": "x^3 + x", "text": "y^2 + (x^3 + x)*y = -x^4 - 6*x^2 + 15" }, "conductor": "1440", "conductor_sort": 1440, "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)", "discriminant_digits": 12, "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" }, "sources": [ "HLP2000" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "HLP2000", "first_source_year": 2000, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 43 (source: \\cite{HLP2000})", "new_group": true, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:37:49 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": 0, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": null, "qsplit": "HLP2000", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 2, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite", "HLP2000" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4) of E₁ = 210e2 (torsion ℤ/2 × ℤ/8) with the Kubert curve E₂ = E_{2,6}(t) (torsion ℤ/2 × ℤ/6), along the bijection of their rational 2-torsion sending 4P₈ to 3P₆. Up to scaling the HLP sextic is C_t: y² = F(x²), with (x, y) ↦ (x², y) onto E₂ and (x, y) ↦ (1/x², y/x³) onto E₁, so J is isogenous over ℚ to E₁ × E₂. Over ℚ(t) the pullbacks of P₈, P₆ and of a 2-torsion point of E₁ mark [2,24]. Here J(𝔽_p) usually has 2-exponent ≥ 16 (over 𝔽_p every Kummer class is matched), so the group structure at three primes is used: every 11-, 13- and 41-integral t ≡ 24 (mod 5863) has J(𝔽₁₁) ≅ [4,48] (2-rank 2), J(𝔽₁₃) ≅ [2,2,48] and J(𝔽₄₁) ≅ [2,2,24,24] (2-exponent 8), so J(ℚ)_tors is exactly [2,24]; the G2-invariants of the members differ (family_2_24_split_certificate.log)", "inherited_from": null } } }, { "key": "2.26", "group": [ 2, 26 ], "bracket": "[2,26]", "label": "ℤ/2 ⊕ ℤ/26", "order": 52, "rank": 2, "classes": { "simple": [ { "id": "2.26.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "0", "2", "3", "-12", "-3", "9" ], "h": [ "1", "0", "1" ], "equation": { "f": "9*x^6 - 3*x^5 - 12*x^4 + 3*x^3 + 2*x^2", "h": "x^2 + 1", "text": "y^2 + (x^2 + 1)*y = 9*x^6 - 3*x^5 - 12*x^4 + 3*x^3 + 2*x^2" }, "conductor": "8136", "conductor_sort": 8136, "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", "discriminant_digits": 14, "lmfdb": { "label": "8136.c.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C0%2C2%2C3%2C-12%2C-3%2C9%5D%2C%5B1%2C0%2C1%5D%5D", "snapshot_label": "8136.c.1" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 49 (source: \\cite{BookerSutherland})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 23 is strict (χ = T^4 + 3*T^3 + 22*T^2 + 69*T + 529)", "verified": "2026-09-18 17:32:48 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "BookerSutherland", "qsplit": null, "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": false, "gsplit": false, "undecided": false }, "exclusions": { "qsplit": { "reason": "A Jacobian split over ℚ has no rational torsion of prime order greater than 7, by Mazur's theorem applied to an elliptic subvariety and its elliptic quotient.", "sources": [ "QSplitPrimeObstruction", "Mazur1977" ], "proof_url": "proofs/qsplit-prime-obstruction.html" }, "gsplit": { "reason": "A geometrically split abelian surface over ℚ has no rational point of order 26 or 39. The proof applies to every geometric splitting degree.", "sources": [ "Geometric2639Obstruction", "Mazur1977", "KenkuMomose1988", "DEvHMZB2021", "DNQuartic2025", "Faltings1983", "SerreTate1968", "Clark2004", "ClarkXarles2008" ], "proof_url": "proofs/geometric-26-39-obstruction.html" } }, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "impossible", "sources": [ "Geometric2639Obstruction", "Mazur1977", "KenkuMomose1988", "DEvHMZB2021", "DNQuartic2025", "Faltings1983", "SerreTate1968", "Clark2004", "ClarkXarles2008" ], "note": "A geometrically split abelian surface over ℚ has no rational point of order 26 or 39. The proof applies to every geometric splitting degree.", "proof_url": "proofs/geometric-26-39-obstruction.html", "inherited_from": null } } }, { "key": "2.28", "group": [ 2, 28 ], "bracket": "[2,28]", "label": "ℤ/2 ⊕ ℤ/28", "order": 56, "rank": 2, "classes": { "simple": [ { "id": "2.28.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "-3", "4", "34", "-42", "-3", "9" ], "h": [ "1", "0", "1" ], "equation": { "f": "9*x^6 - 3*x^5 - 42*x^4 + 34*x^3 + 4*x^2 - 3*x", "h": "x^2 + 1", "text": "y^2 + (x^2 + 1)*y = 9*x^6 - 3*x^5 - 42*x^4 + 34*x^3 + 4*x^2 - 3*x" }, "conductor": "28200", "conductor_sort": 28200, "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", "discriminant_digits": 17, "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" }, "sources": [ "PP2015" ], "rm": false, "discovered_by": "Platonov–Petrunin 2015", "credit_differs": false, "year": null, "historical": false, "first_source": "PP2015", "first_source_year": 2015, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 50 (source: \\cite{BookerSutherland})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 7 is strict (χ = T^4 + T^3 - 2*T^2 + 7*T + 49)", "verified": "2026-09-18 17:32:48 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "PP2015", "qsplit": null, "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": false, "gsplit": false, "undecided": false }, "exclusions": { "gsplit": { "reason": "A principally polarized abelian surface over ℚ that is simple over ℚ and geometrically split has rational 2-primary torsion of order at most four if it has a rational point of order 7. This group has 7-torsion and 2-primary order at least eight, so it is impossible in that class at every geometric splitting degree.", "sources": [ "QSimple228Obstruction", "QSimple2214Obstruction", "QSimple63Obstruction", "Geometric2639Obstruction", "Faltings1983", "KenkuMomose1988" ], "proof_url": "proofs/qsimple-228-obstruction.html" } }, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "open", "sources": [], "note": "", "inherited_from": null } } }, { "key": "2.30", "group": [ 2, 30 ], "bracket": "[2,30]", "label": "ℤ/2 ⊕ ℤ/30", "order": 60, "rank": 2, "classes": { "simple": [], "qsplit": [ { "id": "2.30.a", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "12", "-36", "18", "23", "-13", "-5", "1" ], "h": [ "0", "1", "1" ], "equation": { "f": "x^6 - 5*x^5 - 13*x^4 + 23*x^3 + 18*x^2 - 36*x + 12", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = x^6 - 5*x^5 - 13*x^4 + 23*x^3 + 18*x^2 - 36*x + 12" }, "conductor": "13068", "conductor_sort": 13068, "conductor_note": "N(E1) N(E2) = 66 * 198 for the elliptic curves ['1', '0', '0', '-45', '81'] and ['1', '-1', '1', '-725', '7661'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "discriminant_digits": 16, "lmfdb": { "label": "13068.f.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B12%2C-36%2C18%2C23%2C-13%2C-5%2C1%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "13068.f.2" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 44 (source: \\cite{BookerSutherland})", "new_group": true, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:37:51 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": null, "qsplit": "BookerSutherland", "gsplit": null, "undecided": null }, "known": { "simple": false, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": { "gsplit": { "reason": "A geometrically split abelian surface that is simple over ℚ has no rational point of order 30. The proof applies to every geometric splitting degree and requires simplicity over ℚ.", "sources": [ "QSimple30Obstruction", "DNQuartic2025", "SerreTate1968", "Tate1966", "Mazur1977", "KenkuMomose1988", "ClarkXarles2008", "Clark2004" ], "proof_url": "proofs/qsimple-30-obstruction.html" } }, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite", "HLP2000" ], "note": "the gain locus of the [30] family (the Howe–Leprévost–Poonen (2,2)-gluing of E₁ = 66c1 with E₂ = E₆(u), u = (33s² − 1)/(9 − 33s²)): by HLP 2000, Prop. 12, a half of (T₁, T₁′) (the rational 2-torsion points) glues to a new rational 2-torsion point of J exactly when (β₁ − β₂)(α₁ − α₂) is a square in ℚ(√33). That is the genus-1 curve X: (w₁ + w₂√33)² = c(s − a)(s − b), whose Jacobian 69696gl1 has rank 2; on X, J(ℚ)_tors ⊇ [2,30]. For R_k = T + 8kP on 69696gl1 (P = (385, 7623), T = (22, 0) over s = 3/11), s(R_k) ≡ 3 (mod 5), where J(𝔽₅) ≅ [2,30], so J(ℚ)_tors is exactly [2,30]; the G2-invariants at k = 0, ±1 differ. The census curve 2.30.a is the member s = 3/11 (family_2_30_split_certificate.log)", "inherited_from": null } } }, { "key": "2.48", "group": [ 2, 48 ], "bracket": "[2,48]", "label": "ℤ/2 ⊕ ℤ/48", "order": 96, "rank": 2, "classes": { "simple": [], "qsplit": [ { "id": "2.48.a", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "1", "-13", "19", "111", "19", "-13", "1" ], "h": [ "0", "1", "1" ], "equation": { "f": "x^6 - 13*x^5 + 19*x^4 + 111*x^3 + 19*x^2 - 13*x + 1", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = x^6 - 13*x^5 + 19*x^4 + 111*x^3 + 19*x^2 - 13*x + 1" }, "conductor": "18900", "conductor_sort": 18900, "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)", "discriminant_digits": 21, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 45 (source: \\cite{BookerSutherland})", "new_group": true, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:37:53 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": null, "qsplit": "BookerSutherland", "gsplit": null, "undecided": null }, "known": { "simple": false, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite", "HLP2000" ], "note": "the gain locus of the [2,24] family (the Howe–Leprévost–Poonen (2,2)-gluing of E₁ = 210e2 with E₂ = E_{2,6}(t)): by HLP 2000, Prop. 12, a half of (P₈ + T, T₃′) glues to a rational point of J exactly when 6(t − 5)(t + 3) and 15(t − 1)(t − 9) are both squares. That is the genus-1 curve X whose Jacobian 4800c1 has rank 1; on X the gained point has order 16 and J(ℚ)_tors ⊇ [2,48]. For R_k = T′ + 8kP on 4800c1 (P = (13, −56), T′ = (−3, 0) over t = 21), t(R_k) ≡ 21 (mod 143), where J(𝔽₁₁) ≅ [4,48] and J(𝔽₁₃) ≅ [2,2,48], so J(ℚ)_tors is exactly [2,48]; the G2-invariants at k = 0, ±1 differ. The census curve 2.48.a is the member t = 21 (family_2_48_split_certificate.log)", "inherited_from": null } } }, { "key": "3.3", "group": [ 3, 3 ], "bracket": "[3,3]", "label": "ℤ/3 ⊕ ℤ/3", "order": 9, "rank": 2, "classes": { "simple": [ { "id": "3.3.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "3", "3", "7", "2", "4", "0", "1" ], "h": [ "1" ], "equation": { "f": "x^6 + 4*x^4 + 2*x^3 + 7*x^2 + 3*x + 3", "h": "1", "text": "y^2 + y = x^6 + 4*x^4 + 2*x^3 + 7*x^2 + 3*x + 3" }, "conductor": "5100", "conductor_sort": 5100, "conductor_note": "", "discriminant_digits": 12, "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" }, "sources": [ "BFT2014" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "BFT2014", "first_source_year": 2014, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 51 (source: \\cite{BFT2014})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 19 is strict (χ = T^4 - 4*T^3 + 15*T^2 - 76*T + 361)", "verified": "2026-09-18 17:32:49 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "3.3.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "-2", "6", "-8", "6", "-8", "6", "-2" ], "h": [ "0" ], "equation": { "f": "-2*x^6 + 6*x^5 - 8*x^4 + 6*x^3 - 8*x^2 + 6*x - 2", "h": "0", "text": "y^2 = -2*x^6 + 6*x^5 - 8*x^4 + 6*x^3 - 8*x^2 + 6*x - 2" }, "conductor": "361", "conductor_sort": 361, "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", "discriminant_digits": 15, "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" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 46 (source: \\cite{BSSVY})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:37:54 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [ { "id": "3.3.c", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "196", "0", "0", "-13" ], "h": [ "0", "0", "0", "1" ], "equation": { "f": "-13*x^3 + 196", "h": "x^3", "text": "y^2 + x^3*y = -13*x^3 + 196" }, "conductor": "1764", "conductor_sort": 1764, "conductor_note": "", "discriminant_digits": 17, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": 2026, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "generators", "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)", "new_group": false, "new_for_class": true, "certificate": "the geometric automorphism group has order 12 (7 involutions), so the curve is bielliptic over Qbar; simple over ℚ: χ_43 = T^4 - 4*T^3 - 27*T^2 - 172*T + 1849 is irreducible", "verified": "2026-10-17 06:12:17 UTC", "submitted": "2026-10-16", "source_kind": "file" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": 0, "undecided": null }, "first_source": { "simple": "BFT2014", "qsplit": "BSSVY", "gsplit": "BookerSutherland", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite", "BFT2014" ], "note": "Bruin–Flynn–Testa 2014, Theorem 6, on the line r = 2, s = 3: F = G₁² + 12H₁³ = G₂² + 12tH₂³ with H₁ = x² + 2x + t, H₂ = x² + x + 2; T₁ = [H₁, G₁] and T₂ = [H₂, G₂] have order 3 and T₂ ∉ ⟨T₁⟩, so (ℤ/3)² is marked over ℚ(t). Every t ≡ 19 (mod 65) has #J(𝔽₅) = 9 and #J(𝔽₁₃) = 198, strict at 13 (family_33_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4) of the Hesse-form curve E_t: y² + 3txy + (t³ − 1)y = x³ (P = (0,0) of order 3, E[3] = ⟨P⟩ ⊕ μ₃ with μ₃ over x = −(t² + t + 1)) with its 3-isogenous curve E₁ = E/μ₃, along the isomorphism of 2-torsion induced by the isogeny. A model is C_t: y² = (t³ + 3t² + 3t + 2)x⁶ − (3t⁴ + 15t³ + 45t² + 69t + 30)x⁴ + (6t⁴ + 75t³ + 225t² + 273t + 150)x² − 3t⁴ + 3t³ + 9t² − 15t + 6, with (x, y) ↦ (x², y) onto E and (x, y) ↦ (1/x², y/x³) onto E₁, so J is isogenous over ℚ to E × E₁. Over ℚ(t) the pullbacks of P and of its image in E₁ (both of order 3) mark [3,3]. Every 5-integral t ≡ 2 (mod 5) has J(𝔽₅) ≅ [3,3], of order 9, so J(ℚ)_tors is exactly [3,3]. The census curve 3.3.b is the member t = −8; the G2-invariants of the members differ (family_3_3_split_certificate.log)", "inherited_from": null } } }, { "key": "3.6", "group": [ 3, 6 ], "bracket": "[3,6]", "label": "ℤ/3 ⊕ ℤ/6", "order": 18, "rank": 2, "classes": { "simple": [ { "id": "3.6.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "21", "-2", "7", "-5", "0", "-1" ], "h": [ "1", "1", "1", "1" ], "equation": { "f": "-x^5 - 5*x^3 + 7*x^2 - 2*x + 21", "h": "x^3 + x^2 + x + 1", "text": "y^2 + (x^3 + x^2 + x + 1)*y = -x^5 - 5*x^3 + 7*x^2 - 2*x + 21" }, "conductor": "3780", "conductor_sort": 3780, "conductor_note": "", "discriminant_digits": 14, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 52 (source: \\cite{BookerSutherland})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 13 is strict (χ = T^4 + 2*T^3 + 26*T + 169)", "verified": "2026-09-18 17:32:50 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "3.6.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "27", "-27", "56", "-13", "21", "8", "6" ], "h": [ "0", "1", "1" ], "equation": { "f": "6*x^6 + 8*x^5 + 21*x^4 - 13*x^3 + 56*x^2 - 27*x + 27", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = 6*x^6 + 8*x^5 + 21*x^4 - 13*x^3 + 56*x^2 - 27*x + 27" }, "conductor": "196", "conductor_sort": 196, "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)", "discriminant_digits": 20, "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" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 47 (source: \\cite{BSSVY})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:37:55 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [ { "id": "3.6.c", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "19", "4", "-14", "11", "3", "-5", "2" ], "h": [ "1", "1", "1", "1" ], "equation": { "f": "2*x^6 - 5*x^5 + 3*x^4 + 11*x^3 - 14*x^2 + 4*x + 19", "h": "x^3 + x^2 + x + 1", "text": "y^2 + (x^3 + x^2 + x + 1)*y = 2*x^6 - 5*x^5 + 3*x^4 + 11*x^3 - 14*x^2 + 4*x + 19" }, "conductor": "2700", "conductor_sort": 2700, "conductor_note": "taken from the LMFDB label 2700.b.1 (Magma did not finish the conductor computation within the time limit)", "discriminant_digits": 16, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": 2026, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "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)", "new_group": false, "new_for_class": true, "certificate": "the geometric automorphism group has order 12 (7 involutions), so the curve is bielliptic over Qbar; simple over ℚ: χ_19 = T^4 + 5*T^3 + 6*T^2 + 95*T + 361 is irreducible", "verified": "2026-10-17 06:18:54 UTC", "submitted": "2026-10-16", "source_kind": "file" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": 0, "undecided": null }, "first_source": { "simple": "BookerSutherland", "qsplit": "BSSVY", "gsplit": "BookerSutherland", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "ThisSite", "BFT2014" ], "note": "Bruin–Flynn–Testa's (ℤ/3)² family (BFT 2014, Theorem 6) on the rational curve (r, s, t) = (1 − 3l², −1/(9l²(l−1)²), (3l² − 3l + 1)²), where F = G₁² + 4sH₁³ is a quintic with the rational root x₀ = 2l − 1: T₁ = [H₁, G₁] and T₂ = [H₂, G₂] (order 3, independent) and [x − x₀, 0] (order 2) mark [3,6] over ℚ(l). Every l ≡ 2 (mod 407) has #J(𝔽₁₁) = 126 and #J(𝔽₃₇) = 1746 (gcd 18), strict at 37 (family_3_6_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "a twisted Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4) of the curves of the [6,6] family, E: y² + 3λxy + (λ³ − 1)y = x³ at λ = (2t³ + 1)/(3t²) (P of order 3, T of order 2) and E₁ = E/μ₃, along φ ∘ s, where φ: E → E₁ is the 3-isogeny and s is the automorphism of E[2] fixing T and swapping the other two points of order 2 (Galois-equivariant, as in HLP's Corollary 6); the swap removes the 2-torsion gain. A model is C_t: y² = F(x²) = c(x² − r)F₂(x²) with F₂ quadratic, with (x, y) ↦ (x², y) onto E and (x, y) ↦ (1/x², y/x³) onto E₁, so J is isogenous over ℚ to E × E₁. Over ℚ(t) the pullbacks of P + T (order 6) and of φ(P) (order 3) mark [3,6]. Every t ≡ 234 (mod 935), integral at 5, 11 and 17, has J(𝔽₅) ≅ [6,6], so the odd part of J(ℚ)_tors is the marked (ℤ/3)² and its 2-part is elementary. With F₂ = c₂X² + c₁X + c₀, c₀/c₂ = n² is a square in ℚ(t), and the Galois action on the six roots of f_t is given by the quadratic characters of r, c₁² − 4c₀c₂ and −c₁/c₂ + 2n; their Legendre symbols at 5, 11 and 17 generate (ℤ/2)³, and only the class [x² − r, 0] is invariant, so J(ℚ)[2] = ℤ/2 and J(ℚ)_tors is exactly [3,6]. The census curve 3.6.b is the member t = 1/4; the G2-invariants of the members differ (family_3_6_split_certificate.log)", "inherited_from": null } } }, { "key": "3.9", "group": [ 3, 9 ], "bracket": "[3,9]", "label": "ℤ/3 ⊕ ℤ/9", "order": 27, "rank": 2, "classes": { "simple": [ { "id": "3.9.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "326", "-75", "-230", "-113", "40", "75", "16" ], "h": [ "1", "0", "0", "1" ], "equation": { "f": "16*x^6 + 75*x^5 + 40*x^4 - 113*x^3 - 230*x^2 - 75*x + 326", "h": "x^3 + 1", "text": "y^2 + (x^3 + 1)*y = 16*x^6 + 75*x^5 + 40*x^4 - 113*x^3 - 230*x^2 - 75*x + 326" }, "conductor": "8436", "conductor_sort": 8436, "conductor_note": "", "discriminant_digits": 29, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 53 (source: \\cite{Leprevost1995})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 17 is strict (χ = T^4 - 3*T^3 + 7*T^2 - 51*T + 289)", "verified": "2026-09-18 17:32:51 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "3.9.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "0", "-2", "0", "5", "5", "3", "1" ], "h": [ "1", "1", "1" ], "equation": { "f": "x^6 + 3*x^5 + 5*x^4 + 5*x^3 - 2*x", "h": "x^2 + x + 1", "text": "y^2 + (x^2 + x + 1)*y = x^6 + 3*x^5 + 5*x^4 + 5*x^3 - 2*x" }, "conductor": "2916", "conductor_sort": 2916, "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)", "discriminant_digits": 9, "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" }, "sources": [ "HLP2000" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": true, "year": null, "historical": false, "first_source": "HLP2000", "first_source_year": 2000, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 48 (source: \\cite{HLP2000})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:37:57 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [ { "id": "3.9.c", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "1", "-3", "-1", "7", "6", "4", "2" ], "h": [ "0", "1", "1", "1" ], "equation": { "f": "2*x^6 + 4*x^5 + 6*x^4 + 7*x^3 - x^2 - 3*x + 1", "h": "x^3 + x^2 + x", "text": "y^2 + (x^3 + x^2 + x)*y = 2*x^6 + 4*x^5 + 6*x^4 + 7*x^3 - x^2 - 3*x + 1" }, "conductor": "6804", "conductor_sort": 6804, "conductor_note": "taken from the LMFDB label 6804.a.1 (Magma did not finish the conductor computation within the time limit)", "discriminant_digits": 12, "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" }, "sources": [ "Leprevost1995" ], "rm": false, "discovered_by": "Leprévost 1995", "credit_differs": false, "year": 1995, "historical": false, "first_source": "Leprevost1995", "first_source_year": 1995, "torsion_method": "generators", "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)", "new_group": false, "new_for_class": true, "certificate": "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); simple over ℚ: χ_17 = T^4 + 7*T^2 + 289 is irreducible", "verified": "2026-10-17 06:28:55 UTC", "submitted": "2026-10-16", "source_kind": "file" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": 0, "undecided": null }, "first_source": { "simple": "BookerSutherland", "qsplit": "HLP2000", "gsplit": "Leprevost1995", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the pencil C_t: y² = (t²x³ + 3(t + 1)x + 1)² + 32t²x³, a line of the [9]-chart of 2026-10-13, with the class [x² + (3x + 1)/t, (2t − 12)x − 4] of order 9 over ℚ(t). Since (4t + 1)f_t = B² + 4t²(tx² − x − 1)³ with B = t²x³ + 6t²x² + (9t + 3)x + 2t + 1, after the base change t = (s² − 1)/4 the class [x² − (x + 1)/t, 2s(2x + 1)] is a second rational point of order 3, not in the span of the first, so [3,9] is marked over ℚ(s). Every 7- and 17-integral s ≡ 5 (mod 119) has #J(𝔽₇) = 81 and #J(𝔽₁₇) = 351 (gcd 27), strict at 17 (family_3_9_certificate.log). The census curve 3.9.a is not on the pencil. Leprévost's family C_t (1995), whose Jacobians contain [3,9], is geometrically split for every rational t ≠ 0, 1: it has a degree-3 map onto an elliptic curve over ℚ(√−3) (leprevost_3_9_split.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4) of the Kubert curve E = E₉(t) (torsion ℤ/9, P = (0,0)) with its 3-isogenous curve E₁ = E/⟨3P⟩, along the isomorphism of 2-torsion induced by the isogeny; E₁[3] = ⟨φ(P)⟩ ⊕ μ₃. A model is C_t: y² = F(x²), with (x, y) ↦ (x², y) onto E and (x, y) ↦ (1/x², y/x³) onto E₁, so J is isogenous over ℚ to E × E₁. Over ℚ(t) the pullbacks of P (order 9) and of φ(P) (order 3) mark [3,9]. Here #J(𝔽_p) = #E(𝔽_p)² is always divisible by 81, so the group structure is used instead: every 7-integral t ≡ 2 (mod 7) has J(𝔽₇) ≅ [3,3,9]; J(ℚ)[3] = E(ℚ)[3] ⊕ E₁(ℚ)[3] has rank at most 2, and neither [9,9] nor [3,27] embeds in [3,3,9], so J(ℚ)_tors is exactly [3,9]; the G2-invariants of the members differ (family_3_9_split_certificate.log)", "inherited_from": null } } }, { "key": "3.12", "group": [ 3, 12 ], "bracket": "[3,12]", "label": "ℤ/3 ⊕ ℤ/12", "order": 36, "rank": 2, "classes": { "simple": [ { "id": "3.12.b", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "19644345", "10277505", "231383", "-7653746", "-1261612", "475068", "229392" ], "h": [ "0", "1", "1" ], "equation": { "f": "229392*x^6 + 475068*x^5 - 1261612*x^4 - 7653746*x^3 + 231383*x^2 + 10277505*x + 19644345", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = 229392*x^6 + 475068*x^5 - 1261612*x^4 - 7653746*x^3 + 231383*x^2 + 10277505*x + 19644345" }, "conductor": "951369300", "conductor_sort": 951369300, "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", "discriminant_digits": 75, "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": "" }, "sources": [], "rm": false, "discovered_by": "F. Najman", "credit_differs": false, "year": 2026, "historical": false, "first_source": null, "first_source_year": null, "torsion_method": "generators", "reference": "new", "new_group": false, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 37 is strict (χ = T^4 - T^3 - 37*T + 1369)", "verified": "2026-10-12 18:02:46 UTC", "submitted": "2026-10-12", "source_kind": "issue" } ], "qsplit": [ { "id": "3.12.a", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "-8", "12", "6", "-11", "-3", "3", "1" ], "h": [ "1", "1", "1" ], "equation": { "f": "x^6 + 3*x^5 - 3*x^4 - 11*x^3 + 6*x^2 + 12*x - 8", "h": "x^2 + x + 1", "text": "y^2 + (x^2 + x + 1)*y = x^6 + 3*x^5 - 3*x^4 - 11*x^3 + 6*x^2 + 12*x - 8" }, "conductor": "2700", "conductor_sort": 2700, "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)", "discriminant_digits": 14, "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" }, "sources": [ "HLP2000" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "HLP2000", "first_source_year": 2000, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 49 (source: \\cite{HLP2000})", "new_group": true, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:37:59 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": 0, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": null, "qsplit": "HLP2000", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 2, "infinite": { "simple": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the line (A,B) = (−2/3, −10) of the order-12 chart A(12) in coordinates (A,z,B) = (p², z, pr): the curves C_z have a rational class of order 12, and over the function field of E: w² = (3z²+25)(3z²+121) (conductor 990, rank 1, P = (−15, 30) on the minimal model) a second, independent rational 3-torsion class. For R = 110m·P the curve C_{z(R)} has the same reduction as 3.12.b (z = 5) at 37 and 41, so its torsion divides gcd(#J(𝔽₃₇), #J(𝔽₄₁)) = gcd(1332, 1980) = 36 = #[3,12], and its Frobenius at 37 is strict; the G2-invariants at z = 5 and z = 13/21 differ (family_312_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite", "HLP2000" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4) of E₁ = 90c3 (torsion ℤ/12, non-rational 2-torsion over ℚ(√−15)) with the Kubert curve E₂ = E₆(u), u = (−15s² − 1)/(9 + 15s²) (a line through u = −1 on the conic Δ₁₂w² = (u + 1)(9u + 1) of HLP 2000, §3.5), along the Galois isomorphism of 2-torsion fixing the rational points. Up to scaling the HLP sextic is C_s: y² = F(x²), with (x, y) ↦ (x², y) onto E₂ and (x, y) ↦ (1/x², y/x³) onto E₁, so J is isogenous over ℚ to E₁ × E₂. Over ℚ(s) the pullbacks of the points of order 12 and 6 generate [3,12]. Here #J(𝔽_p) is always divisible by 72, so the group structure is used: every 7- and 11-integral s ≡ −9 (mod 77) has J(𝔽₇) ≅ [12,12] and J(𝔽₁₁) ≅ [3,24] (cyclic 2-part), so J(ℚ)_tors is exactly [3,12]; the G2-invariants of the members differ. The census curve 3.12.a is the member s = 1/3 (family_3_12_split_certificate.log)", "inherited_from": null } } }, { "key": "3.24", "group": [ 3, 24 ], "bracket": "[3,24]", "label": "ℤ/3 ⊕ ℤ/24", "order": 72, "rank": 2, "classes": { "simple": [], "qsplit": [ { "id": "3.24.a", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "4", "6", "-4", "-6", "5", "-3", "1" ], "h": [ "0", "1", "1" ], "equation": { "f": "x^6 - 3*x^5 + 5*x^4 - 6*x^3 - 4*x^2 + 6*x + 4", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = x^6 - 3*x^5 + 5*x^4 - 6*x^3 - 4*x^2 + 6*x + 4" }, "conductor": "8100", "conductor_sort": 8100, "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)", "discriminant_digits": 15, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 50 (source: \\cite{BookerSutherland})", "new_group": true, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:38:00 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": null, "qsplit": "BookerSutherland", "gsplit": null, "undecided": null }, "known": { "simple": false, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite", "HLP2000" ], "note": "the gain locus of the [3,12] family (the Howe–Leprévost–Poonen (2,2)-gluing of E₁ = 90c3 with E₂ = E₆(u), u = (−15s² − 1)/(9 + 15s²)): by HLP 2000, Prop. 12, a half of (P₄, T′) (P₄ of order 4 on E₁, T′ the rational 2-torsion point of E₂) glues to a rational point of J exactly when (β₁ − β₂)(x(P₄) − α₂) is a square in ℚ(√−15). That is the genus-1 curve X: (w₁ + w₂√−15)² = c(s − a)(s − b), whose Jacobian 4800cb1 has rank 1; on X the gained point has order 8 and J(ℚ)_tors ⊇ [3,24]. For R_k = T + 4kP on 4800cb1 (P = (273/16, 5025/64), T = (3, 0) over s = −1), s(R_k) ≡ −1 (mod 7), where J(𝔽₇) ≅ [3,24], so J(ℚ)_tors is exactly [3,24]; the G2-invariants at k = 0, ±1 differ. The census curve 3.24.a is the member s = −1 (family_3_24_split_certificate.log)", "inherited_from": null } } }, { "key": "4.4", "group": [ 4, 4 ], "bracket": "[4,4]", "label": "ℤ/4 ⊕ ℤ/4", "order": 16, "rank": 2, "classes": { "simple": [ { "id": "4.4.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "20493", "6534", "10832", "2601", "1823", "242", "99" ], "h": [ "0", "1", "1" ], "equation": { "f": "99*x^6 + 242*x^5 + 1823*x^4 + 2601*x^3 + 10832*x^2 + 6534*x + 20493", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = 99*x^6 + 242*x^5 + 1823*x^4 + 2601*x^3 + 10832*x^2 + 6534*x + 20493" }, "conductor": "1575", "conductor_sort": 1575, "conductor_note": "", "discriminant_digits": 39, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 54 (source: \\cite{BookerSutherland})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 31 is strict (χ = T^4 + 4*T^3 - 2*T^2 + 124*T + 961)", "verified": "2026-09-18 17:32:52 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "4.4.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "448", "0", "-7", "0", "-2", "0", "7" ], "h": [ "0", "0", "1" ], "equation": { "f": "7*x^6 - 2*x^4 - 7*x^2 + 448", "h": "x^2", "text": "y^2 + x^2*y = 7*x^6 - 2*x^4 - 7*x^2 + 448" }, "conductor": "225", "conductor_sort": 225, "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", "discriminant_digits": 29, "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" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 51 (source: \\cite{BSSVY})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:38:01 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [ { "id": "4.4.c", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "1", "10", "-20", "-40", "735", "-1260", "-540" ], "h": [ "1" ], "equation": { "f": "-540*x^6 - 1260*x^5 + 735*x^4 - 40*x^3 - 20*x^2 + 10*x + 1", "h": "1", "text": "y^2 + y = -540*x^6 - 1260*x^5 + 735*x^4 - 40*x^3 - 20*x^2 + 10*x + 1" }, "conductor": "48384", "conductor_sort": 48384, "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", "discriminant_digits": 29, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": 2026, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "generators", "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)", "new_group": false, "new_for_class": true, "certificate": "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); simple over ℚ: χ_17 = T^4 + 14*T^2 + 289 is irreducible", "verified": "2026-10-17 06:28:57 UTC", "submitted": "2026-10-16", "source_kind": "file" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": 0, "undecided": null }, "first_source": { "simple": "BookerSutherland", "qsplit": "BSSVY", "gsplit": "BookerSutherland", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite" ], "note": "the Richelot quotient of the [2,2,4,4] family by the kernel K = {[(−c²,0) − ∞], [(0,0) − (−b²,0)], [(−a²,0) − (−d²,0)]}, the record's 'partner #5' (the certificate checks s = 5): the pushforwards of the two classes of order 4 mark [4,4] over ℚ(s). Since #J′(𝔽_p) = #J(𝔽_p) is always divisible by 64, group structures are used: every s ≡ 5 (mod 99061) has J′(𝔽₂₃) ≅ [8,64], J′(𝔽₅₉) ≅ [2,2,4,252] and J′(𝔽₇₃) ≅ [8,568]; the only abelian 2-group containing [4,4] that embeds in all three is [4,4], and the odd parts have gcd 1, so J′(ℚ)_tors is exactly [4,4]; strict at 73 (family_44_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4) of the X₁(4) curve E: y² + xy − by = x³ − bx² (P = (0,0) of order 4) at b = t(t − 1)³/(3 − 4t), where E has a rational 3-isogeny (kernel over x = (t − 1)³/(3 − 4t)), with E₁ = E/C₃, along the isomorphism of 2-torsion induced by the isogeny. A model is C_t: y² = F(x²), with (x, y) ↦ (x², y) onto E and (x, y) ↦ (1/x², y/x³) onto E₁, so J is isogenous over ℚ to E × E₁. Over ℚ(t) the pullback of P has order 4, and the gain of the gluing (the Kummer maps commute with the isogeny), computed from a half Q of P over the quartic field ℚ(t)(x(Q)), is a class of order 4 with coefficients in ℚ(t) independent of it, so [4,4] is marked (the pullbacks alone give [2,4]). Every 7-integral t ≡ 2 (mod 7) has J(𝔽₇) ≅ [4,4], of order 16, so J(ℚ)_tors is exactly [4,4]; the G2-invariants of the members differ (family_4_4_split_certificate.log)", "inherited_from": null } } }, { "key": "4.8", "group": [ 4, 8 ], "bracket": "[4,8]", "label": "ℤ/4 ⊕ ℤ/8", "order": 32, "rank": 2, "classes": { "simple": [ { "id": "4.8.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "164", "682", "450", "-435", "272", "-81" ], "h": [ "0", "1", "1" ], "equation": { "f": "-81*x^5 + 272*x^4 - 435*x^3 + 450*x^2 + 682*x + 164", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = -81*x^5 + 272*x^4 - 435*x^3 + 450*x^2 + 682*x + 164" }, "conductor": "913500", "conductor_sort": 913500, "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", "discriminant_digits": 26, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 55 (source: \\cite{BookerSutherland})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 53 is strict (χ = T^4 - 4*T^3 + 62*T^2 - 212*T + 2809)", "verified": "2026-09-18 17:32:53 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "4.8.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "0", "-15", "-5", "-5" ], "h": [ "0", "1", "0", "1" ], "equation": { "f": "-5*x^3 - 5*x^2 - 15*x", "h": "x^3 + x", "text": "y^2 + (x^3 + x)*y = -5*x^3 - 5*x^2 - 15*x" }, "conductor": "225", "conductor_sort": 225, "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", "discriminant_digits": 13, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 52 (source: \\cite{BookerSutherland})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:38:03 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "BookerSutherland", "qsplit": "BookerSutherland", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 2, "infinite": { "simple": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the M₁(8,2^w) odd model y² = xA(x)B(x) of the [4,8] tangent cover, restricted to the rational curve w = (13R − 12)/(12R − 13), R = 3(τ² − 4)/(2(τ² − 9)), on which both 2-descent conditions for halving T₀ = [x, 0] hold identically: P₈ = (−R, y₀) − ∞ (order 8) and the tangent half H of T₀ (order 4) mark [4,8] over ℚ(τ). Every 13- and 41-integral τ ≡ 4 (mod 533) has #J(𝔽₁₃) = 128 and #J(𝔽₄₁) = 1888 (gcd 32), strict at 41 (family_4_8_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4) of E₁ = E_{2,4}(u/4) (P₁ = (0, 0) of order 4) with E₂ = 210e2 (torsion ℤ/2 × ℤ/8, P₂ = (244, 3658)) along an isomorphism of 2-torsion that sends 2P₁ to a 2-torsion point other than 4P₂: C_u: y² = −(81u² + 862u + 81)(81x² − u)(1024x² + (u − 1)²)(700x² + (u + 1)²), with (x, y) ↦ (x², y) onto E₂ and (x, y) ↦ (1/x², y/x³) onto E₁, so J is isogenous over ℚ to E₁ × E₂. The pullbacks of P₁ and P₂ generate [4,8] over ℚ(u) (no gain is needed). Here #J(𝔽_p) is always divisible by 128, so the group structure is used instead: every 11-, 13- and 41-integral u ≡ 3700 (mod 5863) has J(𝔽₁₁) ≅ [2,4,16], J(𝔽₁₃) ≅ [8,16] and J(𝔽₄₁) ≅ [8,8,24], and the largest group embedding in all three is [4,8], so J(ℚ)_tors is exactly [4,8]; the G2-invariants of the members differ (family_4_8_split_certificate.log)", "inherited_from": null } } }, { "key": "4.12", "group": [ 4, 12 ], "bracket": "[4,12]", "label": "ℤ/4 ⊕ ℤ/12", "order": 48, "rank": 2, "classes": { "simple": [ { "id": "4.12.b", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "9", "165", "579", "-1653", "570", "45" ], "h": [ "0", "1", "1" ], "equation": { "f": "45*x^5 + 570*x^4 - 1653*x^3 + 579*x^2 + 165*x + 9", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = 45*x^5 + 570*x^4 - 1653*x^3 + 579*x^2 + 165*x + 9" }, "conductor": "8430300", "conductor_sort": 8430300, "conductor_note": "", "discriminant_digits": 22, "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": "" }, "sources": [], "rm": false, "discovered_by": "F. Najman", "credit_differs": false, "year": 2026, "historical": false, "first_source": null, "first_source_year": null, "torsion_method": "generators", "reference": "new", "new_group": false, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 31 is strict (χ = T^4 + 8*T^3 + 30*T^2 + 248*T + 961)", "verified": "2026-10-13 08:22:34 UTC", "submitted": "2026-10-13", "source_kind": "issue" } ], "qsplit": [ { "id": "4.12.a", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "9", "-27", "53", "-62", "53", "-27", "9" ], "h": [ "0", "1", "1" ], "equation": { "f": "9*x^6 - 27*x^5 + 53*x^4 - 62*x^3 + 53*x^2 - 27*x + 9", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = 9*x^6 - 27*x^5 + 53*x^4 - 62*x^3 + 53*x^2 - 27*x + 9" }, "conductor": "8100", "conductor_sort": 8100, "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)", "discriminant_digits": 16, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 53 (source: \\cite{BookerSutherland})", "new_group": true, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:38:04 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": 0, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": null, "qsplit": "BookerSutherland", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 2, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4) of the Kubert curve E₂ = E₁₂(t) (torsion ℤ/12, P = (0,0)) with its 3-isogenous curve E₁ = E₂/⟨4P⟩, along the isomorphism of 2-torsion induced by the isogeny. Up to a square it is C_t: y² = −(6τ²+6τ+1)((2τ+1)(4τ+1)x² − (6τ+1))q(x)q(−x), q(x) = τ²x² − (6τ²+6τ+1)x + (3τ+1)², τ = t² − t, with (x, y) ↦ (x², y) onto E₂ and (x, y) ↦ (1/x², y/x³) onto E₁, so J is isogenous over ℚ to E₁ × E₂. Over ℚ(t) the pullback of P has order 12, and the gain of the gluing (the Kummer maps commute with the isogeny), computed from a half Q of 3P over the quartic field ℚ(t)(x(Q)), is a class of order 4 with coefficients in ℚ(t) whose double is not in the pullback of ⟨P⟩, so [4,12] is marked. Every 7-integral t ≡ 5 (mod 7) has J(𝔽₇) ≅ [12,12]; the kernel of E₁ → E₂ is μ₃ and 7 ≡ 1 (mod 3), so a rational point of order 3 on E₁ would give J(𝔽₇)[3] rank ≥ 3, and J(ℚ)_tors is exactly [4,12]. The census curve 4.12.a is the member t = 1/3; the G2-invariants of the members differ (family_412_split_certificate.log)", "inherited_from": null } } }, { "key": "4.16", "group": [ 4, 16 ], "bracket": "[4,16]", "label": "ℤ/4 ⊕ ℤ/16", "order": 64, "rank": 2, "classes": { "simple": [], "qsplit": [ { "id": "4.16.a", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "252", "0", "-24", "0", "-4" ], "h": [ "0", "1", "0", "1" ], "equation": { "f": "-4*x^4 - 24*x^2 + 252", "h": "x^3 + x", "text": "y^2 + (x^3 + x)*y = -4*x^4 - 24*x^2 + 252" }, "conductor": "50400", "conductor_sort": 50400, "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)", "discriminant_digits": 20, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 54 (source: \\cite{BookerSutherland})", "new_group": true, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:38:06 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": null, "qsplit": "BookerSutherland", "gsplit": null, "undecided": null }, "known": { "simple": false, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "Split4162026", "HLP2000" ], "note": "For every s ∈ ℚ, put t = −[3(3s² − 10s + 5)² + (3s² − 5)²]/[4(3s² − 5)²] and D = 256t² + 47t + 16. The curve y² = D(16x² + 9t)(175x² − 9(4t − 1)²)(9x² + (4t + 1)²) is smooth of genus 2, its Jacobian splits over ℚ, and its rational torsion is exactly [4,16]. It is the HLP (2,2)-gluing of E_t = [1, −(t² − 1/16), −(t² − 1/16), 0, 0] and F = [47/42, 5/49, 5/49, 0, 0], with exact elliptic torsion [2,4] and [2,8]. Their torsion Kummer images intersect in precisely the identity and (−15, −6, 10), forcing Jacobian torsion order 64; the matching class gives an order-16 point and group [4,16]. The nonconstant elliptic quotient j-invariant proves infinitely many geometric isomorphism classes. The member s = 1 (t = −1) is the census curve 4.16.a. No congruence restrictions or exceptional thin set are needed (complete proof and family_416_split_certificate.log, 2026-10-20).", "inherited_from": null } } }, { "key": "5.5", "group": [ 5, 5 ], "bracket": "[5,5]", "label": "ℤ/5 ⊕ ℤ/5", "order": 25, "rank": 2, "classes": { "simple": [ { "id": "5.5.b", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "428", "-264", "295", "75", "121", "45", "4" ], "h": [ "0", "0", "0", "1" ], "equation": { "f": "4*x^6 + 45*x^5 + 121*x^4 + 75*x^3 + 295*x^2 - 264*x + 428", "h": "x^3", "text": "y^2 + x^3*y = 4*x^6 + 45*x^5 + 121*x^4 + 75*x^3 + 295*x^2 - 264*x + 428" }, "conductor": "1275918336", "conductor_sort": 1275918336, "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", "discriminant_digits": 28, "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": "" }, "sources": [], "rm": false, "discovered_by": "F. Najman", "credit_differs": false, "year": 2026, "historical": false, "first_source": null, "first_source_year": null, "torsion_method": "generators", "reference": "new", "new_group": false, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 23 is strict (χ = T^4 + 2*T^3 - 3*T^2 + 46*T + 529)", "verified": "2026-10-12 15:04:49 UTC", "submitted": "2026-10-12", "source_kind": "issue" } ], "qsplit": [ { "id": "5.5.a", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "-2", "4", "2", "5", "2", "1" ], "h": [ "1", "1", "1", "1" ], "equation": { "f": "x^5 + 2*x^4 + 5*x^3 + 2*x^2 + 4*x - 2", "h": "x^3 + x^2 + x + 1", "text": "y^2 + (x^3 + x^2 + x + 1)*y = x^5 + 2*x^4 + 5*x^3 + 2*x^2 + 4*x - 2" }, "conductor": "121", "conductor_sort": 121, "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", "discriminant_digits": 12, "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" }, "sources": [ "HLP2000" ], "rm": false, "discovered_by": "Howe–Leprévost–Poonen 2000", "credit_differs": false, "year": null, "historical": false, "first_source": "HLP2000", "first_source_year": 2000, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 55 (source: \\cite{HLP2000})", "new_group": true, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:38:07 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": 0, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": null, "qsplit": "HLP2000", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": { "gsplit": { "reason": "If a geometrically split abelian surface over ℚ is simple over ℚ, its rational p-torsion has rank at most 1 for every prime p ≥ 5. The proof applies to every geometric splitting degree and requires simplicity over ℚ.", "sources": [ "QSimpleRankTwoObstruction", "Beauville2010", "MilneAV2008" ], "proof_url": "proofs/qsimple-rank-two-obstruction.html" } }, "n_curves": 2, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite", "HLP2000" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4) of the X₁(5) curve E: y² + (1 − s)xy − sy = x³ − sx² (P = (0,0) of order 5) at s = t⁵ with its 5-isogenous curve E₁ = E/⟨P⟩, along the isomorphism of 2-torsion induced by the isogeny. The kernel of E₁ → E is μ₅, and for s = t⁵ the curve E₁ has a rational point Q₁ of order 5, so E₁[5] = ⟨Q₁⟩ ⊕ μ₅. A model is C_t: y² = −s²(s² + 1)x⁶ + (s⁶ − 6s⁵ − 6s⁴ − 6s² + 6s + 1)x⁴ − (2s⁶ + 36s⁵ − 15s⁴ − 15s² − 36s + 2)x² + s⁶ − 22s⁵ + 120s⁴ + 120s² + 22s + 1 with s = t⁵, with (x, y) ↦ (x², y) onto E and (x, y) ↦ (1/x², y/x³) onto E₁, so J is isogenous over ℚ to E × E₁; at t = 1 it is HLP's curve y² = −2x⁶ − 10x⁴ + 26x² + 242 (X₀(22), Corollary 5), the census curve 5.5.a. Over ℚ(t) the pullbacks of P and Q₁ mark [5,5]. Every 3-integral t ≡ 1 (mod 3) has J(𝔽₃) ≅ [5,5], of order 25, so J(ℚ)_tors is exactly [5,5]; the G2-invariants of the members differ (family_5_5_split_certificate.log)", "inherited_from": null } } }, { "key": "5.10", "group": [ 5, 10 ], "bracket": "[5,10]", "label": "ℤ/5 ⊕ ℤ/10", "order": 50, "rank": 2, "classes": { "simple": [], "qsplit": [ { "id": "5.10.a", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "-135", "135", "11", "-33", "15", "-3", "9" ], "h": [ "0", "1", "1" ], "equation": { "f": "9*x^6 - 3*x^5 + 15*x^4 - 33*x^3 + 11*x^2 + 135*x - 135", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = 9*x^6 - 3*x^5 + 15*x^4 - 33*x^3 + 11*x^2 + 135*x - 135" }, "conductor": "85500", "conductor_sort": 85500, "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)", "discriminant_digits": 27, "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" }, "sources": [ "HLP2000" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "HLP2000", "first_source_year": 2000, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 56 (source: \\cite{HLP2000})", "new_group": true, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:38:09 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": null, "qsplit": "HLP2000", "gsplit": null, "undecided": null }, "known": { "simple": false, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": { "gsplit": { "reason": "If a geometrically split abelian surface over ℚ is simple over ℚ, its rational p-torsion has rank at most 1 for every prime p ≥ 5. The proof applies to every geometric splitting degree and requires simplicity over ℚ.", "sources": [ "QSimpleRankTwoObstruction", "Beauville2010", "MilneAV2008" ], "proof_url": "proofs/qsimple-rank-two-obstruction.html" } }, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite", "HLP2000" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4 and §3.5) of E₁ = E₁₀(t) with E₂ = E₁₀(u), u = (2t⁴ − 2t + 1)/(8t³ − 8t² + 1), along the isomorphism of 2-torsion that matches the rational points of order 2: (t, u) is the section 2σ of HLP's elliptic surface g(t)y² = g(u), g = (2u − 1)(4u² − 2u − 1), σ = (t, −1), zero section (t, 1), so g(t)g(u) is a square. Up to a square C_t: y² = F(x²), with (x, y) ↦ (x², y) onto E₂ and (x, y) ↦ (1/x², y/x³) onto E₁, so J is isogenous over ℚ to E₁ × E₂. Over ℚ(t) the pullbacks D, D′ of the points of order 10 satisfy 5D = 5D′ and 2D′ ∉ ⟨2D⟩, so [5,10] is marked. Here #J(𝔽_p) is always divisible by 100, so the group structure is used: every 7- and 13-integral t ≡ −1 (mod 91) has the reductions of t = −1 at 7 and 13, where J(𝔽₇) ≅ [10,10] and J(𝔽₁₃) ≅ [5,40], so J(ℚ)_tors is exactly [5,10]; the G2-invariants of the members differ (family_5_10_split_certificate.log)", "inherited_from": null } } }, { "key": "6.6", "group": [ 6, 6 ], "bracket": "[6,6]", "label": "ℤ/6 ⊕ ℤ/6", "order": 36, "rank": 2, "classes": { "simple": [ { "id": "6.6.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "4875", "-1691", "6969", "-3000", "1872" ], "h": [ "0" ], "equation": { "f": "1872*x^5 - 3000*x^4 + 6969*x^3 - 1691*x^2 + 4875*x", "h": "0", "text": "y^2 = 1872*x^5 - 3000*x^4 + 6969*x^3 - 1691*x^2 + 4875*x" }, "conductor": "17506967961600", "conductor_sort": 17506967961600, "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", "discriminant_digits": 36, "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": "" }, "sources": [ "BNSS2026" ], "rm": false, "discovered_by": "Balakrishnan–Najman–Shnidman–Sutherland 2026", "credit_differs": false, "year": null, "historical": false, "first_source": "BNSS2026", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 56 (source: new)", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 37 is strict (χ = T^4 - 10*T^3 + 54*T^2 - 370*T + 1369)", "verified": "2026-09-18 17:32:54 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "6.6.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "1", "3", "6", "7", "6", "3", "1" ], "h": [ "0", "1", "1" ], "equation": { "f": "x^6 + 3*x^5 + 6*x^4 + 7*x^3 + 6*x^2 + 3*x + 1", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = x^6 + 3*x^5 + 6*x^4 + 7*x^3 + 6*x^2 + 3*x + 1" }, "conductor": "196", "conductor_sort": 196, "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)", "discriminant_digits": 8, "lmfdb": { "label": "196.a.21952.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/196/a/21952/1", "snapshot_label": "" }, "sources": [ "HLP2000" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": true, "year": null, "historical": false, "first_source": "HLP2000", "first_source_year": 2000, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 57 (source: \\cite{HLP2000})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:38:11 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "BNSS2026", "qsplit": "HLP2000", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 2, "infinite": { "simple": { "grade": "open", "sources": [ "Najman2026" ], "note": "two isolated geometrically simple examples are known (one on the contact-6 × cubic-contact locus, one on the A(12)-type chart of 2026-10-13); no rational or positive-rank curve through them is known (the contact-6 × contact-6 locus itself is split)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4) of the Hesse-form curve E: y² + 3λxy + (λ³ − 1)y = x³ at λ = (2t³ + 1)/(3t²), where E has the point P = (0,0) of order 3, a point T of order 2 with x(T) = −((t³ − 1)/(3t²))² and E[3] = ⟨P⟩ ⊕ μ₃, with its 3-isogenous curve E₁ = E/μ₃, along the isomorphism of 2-torsion induced by the isogeny. A model is C_t: y² = F(x²) = c(x² − r)q(x)q(−x) over ℚ(t), with (x, y) ↦ (x², y) onto E and (x, y) ↦ (1/x², y/x³) onto E₁, so J is isogenous over ℚ to E × E₁. Over ℚ(t) the pullbacks of P + T and of its image in E₁ (both of order 6) and the class [q, 0] (the gain of the gluing: the Kummer maps commute with the isogeny) mark [6,6]. Every 5-integral t ≡ 3 (mod 5) has J(𝔽₅) ≅ [6,6], of order 36, so J(ℚ)_tors is exactly [6,6]. The census curve 6.6.b is the member t = −2; the G2-invariants of the members differ (family_6_6_split_certificate.log)", "inherited_from": null } } }, { "key": "6.12", "group": [ 6, 12 ], "bracket": "[6,12]", "label": "ℤ/6 ⊕ ℤ/12", "order": 72, "rank": 2, "classes": { "simple": [], "qsplit": [ { "id": "6.12.a", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "88825", "81950", "75207", "-13354", "-6347", "396", "132" ], "h": [ "0" ], "equation": { "f": "132*x^6 + 396*x^5 - 6347*x^4 - 13354*x^3 + 75207*x^2 + 81950*x + 88825", "h": "0", "text": "y^2 = 132*x^6 + 396*x^5 - 6347*x^4 - 13354*x^3 + 75207*x^2 + 81950*x + 88825" }, "conductor": "18900", "conductor_sort": 18900, "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)", "discriminant_digits": 46, "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": "" }, "sources": [ "HLP2000" ], "rm": false, "discovered_by": "Balakrishnan–Najman–Shnidman–Sutherland 2026", "credit_differs": true, "year": null, "historical": false, "first_source": "HLP2000", "first_source_year": 2000, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 58 (source: \\cite{HLP2000})", "new_group": true, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:38:12 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": null, "qsplit": "HLP2000", "gsplit": null, "undecided": null }, "known": { "simple": false, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite", "HLP2000" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4 and §3.5) of E₁ = E₁₂(t) with E₂ = E₁₂(u) along the isomorphism of 2-torsion that matches the rational points of order 2, where (t, u) is the section 2σ of HLP's elliptic surface q(t)y² = q(u), q = (2u² − 2u + 1)(6u² − 6u + 1), σ = (t, −1), zero section (t, 1) (u of degree 9 in t; u = t and u = 1 − t give E₂ ≅ E₁). Up to a square C_t: y² = F(x²), with (x, y) ↦ (x², y) onto E₂ and (x, y) ↦ (1/x², y/x³) onto E₁, so J is isogenous over ℚ to E₁ × E₂. Over ℚ(t) the pullbacks D, D′ of the points of order 12 satisfy 6D = 6D′, and D′ − D has order 6 and meets ⟨D⟩ trivially, so [6,12] is marked. Here #J(𝔽_p) is always divisible by 144, so the group structure is used: every 7- and 17-integral t ≡ 3 (mod 119) has the reductions of t = 3 at 7 and 17, where J(𝔽₇) ≅ [12,12] and J(𝔽₁₇) ≅ [6,48], so J(ℚ)_tors is exactly [6,12]; the G2-invariants of the members differ (family_6_12_split_certificate.log)", "inherited_from": null } } }, { "key": "7.7", "group": [ 7, 7 ], "bracket": "[7,7]", "label": "ℤ/7 ⊕ ℤ/7", "order": 49, "rank": 2, "classes": { "simple": [], "qsplit": [ { "id": "7.7.a", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "869675859", "0", "3232987", "0", "3025", "0", "1" ], "h": [ "0" ], "equation": { "f": "x^6 + 3025*x^4 + 3232987*x^2 + 869675859", "h": "0", "text": "y^2 = x^6 + 3025*x^4 + 3232987*x^2 + 869675859" }, "conductor": "53958996", "conductor_sort": 53958996, "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", "discriminant_digits": 54, "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": "" }, "sources": [ "HLP2000" ], "rm": false, "discovered_by": "Howe–Leprévost–Poonen 2000", "credit_differs": false, "year": null, "historical": false, "first_source": "HLP2000", "first_source_year": 2000, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 59 (source: \\cite{HLP2000})", "new_group": true, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:38:14 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": null, "qsplit": "HLP2000", "gsplit": null, "undecided": null }, "known": { "simple": false, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": { "gsplit": { "reason": "If a geometrically split abelian surface over ℚ is simple over ℚ, its rational p-torsion has rank at most 1 for every prime p ≥ 5. The proof applies to every geometric splitting degree and requires simplicity over ℚ.", "sources": [ "QSimpleRankTwoObstruction", "Beauville2010", "MilneAV2008" ], "proof_url": "proofs/qsimple-rank-two-obstruction.html" } }, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "open", "sources": [ "HLP2000" ], "note": "HLP 2000, Table 1: a single curve (parameter space ℙ⁰)", "inherited_from": null } } }, { "key": "8.8", "group": [ 8, 8 ], "bracket": "[8,8]", "label": "ℤ/8 ⊕ ℤ/8", "order": 64, "rank": 2, "classes": { "simple": [], "qsplit": [ { "id": "8.8.a", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "84285504", "-4535664", "-4045487", "1800118", "88597", "88596", "836" ], "h": [ "0" ], "equation": { "f": "836*x^6 + 88596*x^5 + 88597*x^4 + 1800118*x^3 - 4045487*x^2 - 4535664*x + 84285504", "h": "0", "text": "y^2 = 836*x^6 + 88596*x^5 + 88597*x^4 + 1800118*x^3 - 4045487*x^2 - 4535664*x + 84285504" }, "conductor": "17243100", "conductor_sort": 17243100, "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)", "discriminant_digits": 72, "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": "" }, "sources": [ "HLP2000" ], "rm": false, "discovered_by": "Balakrishnan–Najman–Shnidman–Sutherland 2026", "credit_differs": true, "year": null, "historical": false, "first_source": "HLP2000", "first_source_year": 2000, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 60 (source: \\cite{HLP2000})", "new_group": true, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:38:16 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": null, "qsplit": "HLP2000", "gsplit": null, "undecided": null }, "known": { "simple": false, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite", "Genus2TorsionNotebook" ], "note": "the conjugation self-gluing of the X₁(8) curve E_t at t = m²/(m² + 1) (Genus2TorsionNotebook): with N = m⁸ − 4m⁶ − 2m⁴ − 4m² + 1 and F(u) = −N(u + 1)(16m⁸(u + 1)² − N²u), C_m: y² = F(x²); (x, y) ↦ (x², y) and (x, y) ↦ (1/x², y/x³) map C_m onto E′: Y² = F(u) ≅ E_t, so J(C_m) is isogenous over ℚ to E′ × E′. [8,8] is marked over ℚ(m). Here #J(𝔽_p) = #E′(𝔽_p)² is always divisible by 256, so the group structure is used instead: every m ≡ 2 (mod 4381) has the reductions of m = 2 at 13 and 337, with J(𝔽₁₃) = (ℤ/16)² (no odd part, 2-rank 2) and J(𝔽₃₃₇) = [8,8,40,40] (2-primary exponent 8), so J(ℚ)_tors is exactly [8,8] (family_88_split_certificate.log)", "inherited_from": null } } }, { "key": "2.2.2", "group": [ 2, 2, 2 ], "bracket": "[2,2,2]", "label": "ℤ/2 ⊕ ℤ/2 ⊕ ℤ/2", "order": 8, "rank": 3, "classes": { "simple": [ { "id": "2.2.2.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "1", "-3", "-5", "8", "10" ], "h": [ "0", "1" ], "equation": { "f": "10*x^5 + 8*x^4 - 5*x^3 - 3*x^2 + x", "h": "x", "text": "y^2 + x*y = 10*x^5 + 8*x^4 - 5*x^3 - 3*x^2 + x" }, "conductor": "2600", "conductor_sort": 2600, "conductor_note": "", "discriminant_digits": 6, "lmfdb": { "label": "2600.a.338000.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/2600/a/338000/1", "snapshot_label": "" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 57 (source: \\cite{BSSVY})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 11 is strict (χ = T^4 + 4*T^3 + 6*T^2 + 44*T + 121)", "verified": "2026-09-18 17:32:55 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "2.2.2.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "0", "1", "32", "268", "129", "16" ], "h": [ "0", "1" ], "equation": { "f": "16*x^5 + 129*x^4 + 268*x^3 + 32*x^2 + x", "h": "x", "text": "y^2 + x*y = 16*x^5 + 129*x^4 + 268*x^3 + 32*x^2 + x" }, "conductor": "315", "conductor_sort": 315, "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", "discriminant_digits": 9, "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" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 61 (source: \\cite{BSSVY})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:38:17 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [ { "id": "2.2.2.c", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "0", "-16", "-8", "23", "-10", "-25" ], "h": [ "0" ], "equation": { "f": "-25*x^5 - 10*x^4 + 23*x^3 - 8*x^2 - 16*x", "h": "0", "text": "y^2 = -25*x^5 - 10*x^4 + 23*x^3 - 8*x^2 - 16*x" }, "conductor": "775", "conductor_sort": 775, "conductor_note": "taken from the LMFDB label 775.a.2 (Magma did not finish the conductor computation within the time limit)", "discriminant_digits": 17, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": 2026, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "generators", "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)", "new_group": false, "new_for_class": true, "certificate": "(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); simple over ℚ: χ_3 = T^4 - 2*T^2 + 9 is irreducible", "verified": "2026-10-17 06:11:58 UTC", "submitted": "2026-10-16", "source_kind": "file" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": 0, "undecided": null }, "first_source": { "simple": "BSSVY", "qsplit": "BSSVY", "gsplit": "BookerSutherland", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite" ], "note": "y² = x(x−1)(x−2)(x² + x + t): [x, 0], [x−1, 0], [x−2, 0] mark [2,2,2] over ℚ(t). Every t ≡ 3 (mod 581) has #J(𝔽₇) = 32 and #J(𝔽₈₃) = 6824 (gcd 8), strict at 83 (family_2_2_2_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings (Bröker–Howe–Lauter–Stevenhagen normal form) of E = 17a2 (torsion ℤ/2 × ℤ/2) with the partners F that have a rational 2-torsion point. The partners anti-symplectically 3-congruent to E form a rational curve, and so does the cover on which F has a rational 2-torsion point (Γ(3) ∩ Γ₀(2) has genus 0); this gives C_v over ℚ(v) with explicit degree-3 maps onto E and F_v, so J is isogenous over ℚ to E × F_v. The map onto E has prime degree, so pullback is injective on E and [2,2] ⊂ J(ℚ); over ℚ(v) the quadratic divisors of f_v give J(ℚ)[2] of 2-rank 3, one more, so [2,2,2] is marked. Every 7- and 19-integral v ≡ −1 (mod 133) has the reductions of v = −1 at 7 and 19, where J(𝔽₇) ≅ [2,2,10] and J(𝔽₁₉) ≅ [2,2,108]; the largest group embedding in both is [2,2,2], so J(ℚ)_tors is exactly [2,2,2]; the G2-invariants at v = −1 and v = 132 differ (family_2_2_2_split_certificate.log)", "inherited_from": null } } }, { "key": "2.2.4", "group": [ 2, 2, 4 ], "bracket": "[2,2,4]", "label": "ℤ/2 ⊕ ℤ/2 ⊕ ℤ/4", "order": 16, "rank": 3, "classes": { "simple": [ { "id": "2.2.4.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "6", "-8", "-3", "3", "1" ], "h": [ "0", "1", "1" ], "equation": { "f": "x^5 + 3*x^4 - 3*x^3 - 8*x^2 + 6*x", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = x^5 + 3*x^4 - 3*x^3 - 8*x^2 + 6*x" }, "conductor": "3978", "conductor_sort": 3978, "conductor_note": "", "discriminant_digits": 9, "lmfdb": { "label": "3978.a.930852.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/3978/a/930852/1", "snapshot_label": "" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 58 (source: \\cite{BSSVY})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 23 is strict (χ = T^4 - 2*T^3 + 30*T^2 - 46*T + 529)", "verified": "2026-09-18 17:32:55 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "2.2.4.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "0", "-15", "7", "18", "0", "3", "-2" ], "h": [ "0", "1", "0", "1" ], "equation": { "f": "-2*x^6 + 3*x^5 + 18*x^3 + 7*x^2 - 15*x", "h": "x^3 + x", "text": "y^2 + (x^3 + x)*y = -2*x^6 + 3*x^5 + 18*x^3 + 7*x^2 - 15*x" }, "conductor": "225", "conductor_sort": 225, "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", "discriminant_digits": 19, "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" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 62 (source: \\cite{BSSVY})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:40:49 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [ { "id": "2.2.4.c", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "19", "71", "67", "-8", "-19", "4" ], "h": [ "1", "1" ], "equation": { "f": "4*x^5 - 19*x^4 - 8*x^3 + 67*x^2 + 71*x + 19", "h": "x + 1", "text": "y^2 + (x + 1)*y = 4*x^5 - 19*x^4 - 8*x^3 + 67*x^2 + 71*x + 19" }, "conductor": "4752", "conductor_sort": 4752, "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", "discriminant_digits": 12, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": 2026, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "generators", "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)", "new_group": false, "new_for_class": true, "certificate": "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); simple over ℚ: χ_19 = T^4 - 10*T^2 + 361 is irreducible", "verified": "2026-10-17 06:12:03 UTC", "submitted": "2026-10-16", "source_kind": "file" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": 0, "undecided": null }, "first_source": { "simple": "BSSVY", "qsplit": "BSSVY", "gsplit": "BookerSutherland", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite" ], "note": "y² = x(x+4)(x+9)(x² + (2t+1)x + t²): D = [x² + (t−1)x − 6t, x(4x + 5t + 6) mod u] has 2D = [x, 0] ≠ 0, and with [x+4, 0], [x+9, 0] it marks [2,2,4] over ℚ(t). Every t ≡ 1 (mod 91) has #J(𝔽₇) = 32 and #J(𝔽₁₃) = 208 (gcd 16), strict at 13 (family_224_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings (Bröker–Howe–Lauter–Stevenhagen normal form) of E = 15a1 (torsion ℤ/2 × ℤ/4) with the partners F that have a rational 2-torsion point. The partners anti-symplectically 3-congruent to E form a rational curve, and so does the cover on which F has a rational 2-torsion point (Γ(3) ∩ Γ₀(2) has genus 0); this gives C_v over ℚ(v) with explicit degree-3 maps onto E and F_v, so J is isogenous over ℚ to E × F_v. The map onto E has prime degree, so pullback is injective on E and [2,4] ⊂ J(ℚ); over ℚ(v) the quadratic divisors of f_v give J(ℚ)[2] of 2-rank 3, one more, so [2,2,4] is marked. Every 11- and 17-integral v ≡ 4 (mod 187) has the reductions of v = 4 at 11 and 17, where J(𝔽₁₁) ≅ [2,2,40] and J(𝔽₁₇) ≅ [2,4,44]; the largest group embedding in both is [2,2,4], so J(ℚ)_tors is exactly [2,2,4]; the G2-invariants at v = 4 and v = −183 differ (family_2_2_4_split_certificate.log)", "inherited_from": null } } }, { "key": "2.2.6", "group": [ 2, 2, 6 ], "bracket": "[2,2,6]", "label": "ℤ/2 ⊕ ℤ/2 ⊕ ℤ/6", "order": 24, "rank": 3, "classes": { "simple": [ { "id": "2.2.6.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "1", "-1", "-4", "0", "3" ], "h": [ "1", "0", "1" ], "equation": { "f": "3*x^5 - 4*x^3 - x^2 + x", "h": "x^2 + 1", "text": "y^2 + (x^2 + 1)*y = 3*x^5 - 4*x^3 - x^2 + x" }, "conductor": "816", "conductor_sort": 816, "conductor_note": "", "discriminant_digits": 7, "lmfdb": { "label": "816.a.39168.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/816/a/39168/1", "snapshot_label": "" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 59 (source: \\cite{BSSVY})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 29 is strict (χ = T^4 - 8*T^3 + 46*T^2 - 232*T + 841)", "verified": "2026-09-18 17:32:56 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "2.2.6.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "0", "1", "-4", "-3", "3", "2" ], "h": [ "0", "1", "0", "1" ], "equation": { "f": "2*x^5 + 3*x^4 - 3*x^3 - 4*x^2 + x", "h": "x^3 + x", "text": "y^2 + (x^3 + x)*y = 2*x^5 + 3*x^4 - 3*x^3 - 4*x^2 + x" }, "conductor": "600", "conductor_sort": 600, "conductor_note": "", "discriminant_digits": 8, "lmfdb": { "label": "600.a.18000.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/600/a/18000/1", "snapshot_label": "" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 63 (source: \\cite{BSSVY})", "new_group": false, "new_for_class": true, "certificate": "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]", "verified": "2026-09-18 17:40:50 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [ { "id": "2.2.6.c", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "1", "-2", "-7", "10", "14", "-8", "-8" ], "h": [ "0" ], "equation": { "f": "-8*x^6 - 8*x^5 + 14*x^4 + 10*x^3 - 7*x^2 - 2*x + 1", "h": "0", "text": "y^2 = -8*x^6 - 8*x^5 + 14*x^4 + 10*x^3 - 7*x^2 - 2*x + 1" }, "conductor": "6912", "conductor_sort": 6912, "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", "discriminant_digits": 14, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": 2026, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "generators", "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)", "new_group": false, "new_for_class": true, "certificate": "(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); simple over ℚ: χ_5 = T^4 - 2*T^2 + 25 is irreducible", "verified": "2026-10-17 06:12:05 UTC", "submitted": "2026-10-16", "source_kind": "file" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": 0, "undecided": null }, "first_source": { "simple": "BSSVY", "qsplit": "BSSVY", "gsplit": "BookerSutherland", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "ThisSite" ], "note": "y² = F₁F₂ with F₁ = x(x² − (t+3)x + 3t + 2), F₂ = (x−1)(x−2)(x−t): F₁ − F₂ = 2t is constant, so D₃ = ∞₊ − ∞₋ has order 3, and the four rational roots give rational 2-rank 3: [2,2,6] is marked over ℚ(t). Every t ≡ −2 (mod 145) has #J(𝔽₅) = 24 and #J(𝔽₂₉) = 648, strict at 29 (family_2_2_6_certificate.log); t = −2 is the census curve 2.2.6.a", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings (Bröker–Howe–Lauter–Stevenhagen normal form) of E = 30a2 (torsion ℤ/2 × ℤ/6) with the partners F that have a rational 2-torsion point. The partners anti-symplectically 3-congruent to E form a rational curve, and so does the cover on which F has a rational 2-torsion point (Γ(3) ∩ Γ₀(2) has genus 0); this gives C_v over ℚ(v) with explicit degree-3 maps onto E and F_v, so J is isogenous over ℚ to E × F_v. The map onto E has prime degree, so pullback is injective on E and [2,6] ⊂ J(ℚ); over ℚ(v) the quadratic divisors of f_v give J(ℚ)[2] of 2-rank 3, one more, so [2,2,6] is marked. Every 13- and 17-integral v ≡ 3 (mod 221) has the reductions of v = 3 at 13 and 17, where J(𝔽₁₃) ≅ [2,6,12] and J(𝔽₁₇) ≅ [2,2,54]; the largest group embedding in both is [2,2,6], so J(ℚ)_tors is exactly [2,2,6]; the G2-invariants at v = 3 and v = −218 differ. The member v = −1 (outside the class) is the census curve 2.2.6.b (family_2_2_6_split_certificate.log)", "inherited_from": null } } }, { "key": "2.2.8", "group": [ 2, 2, 8 ], "bracket": "[2,2,8]", "label": "ℤ/2 ⊕ ℤ/2 ⊕ ℤ/8", "order": 32, "rank": 3, "classes": { "simple": [ { "id": "2.2.8.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "-1", "-6", "-3", "20", "9" ], "h": [ "0", "1", "1" ], "equation": { "f": "9*x^5 + 20*x^4 - 3*x^3 - 6*x^2 - x", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = 9*x^5 + 20*x^4 - 3*x^3 - 6*x^2 - x" }, "conductor": "3942", "conductor_sort": 3942, "conductor_note": "", "discriminant_digits": 11, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 60 (source: \\cite{BookerSutherland})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 7 is strict (χ = T^4 + 2*T^3 - 2*T^2 + 14*T + 49)", "verified": "2026-09-18 17:32:57 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "2.2.8.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "0", "1", "1", "-5", "-4", "6" ], "h": [ "1", "1" ], "equation": { "f": "6*x^5 - 4*x^4 - 5*x^3 + x^2 + x", "h": "x + 1", "text": "y^2 + (x + 1)*y = 6*x^5 - 4*x^4 - 5*x^3 + x^2 + x" }, "conductor": "360", "conductor_sort": 360, "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)", "discriminant_digits": 5, "lmfdb": { "label": "360.a.6480.1", "kind": "production", "url": "https://www.lmfdb.org/Genus2Curve/Q/360/a/6480/1", "snapshot_label": "" }, "sources": [ "BSSVY" ], "rm": false, "discovered_by": "LMFDB genus 2 database (BSSVY 2016)", "credit_differs": false, "year": null, "historical": false, "first_source": "BSSVY", "first_source_year": 2016, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 64 (source: \\cite{BSSVY})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:40:51 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [ { "id": "2.2.8.c", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "0", "-25", "46", "5", "-39", "3", "9" ], "h": [ "0", "1", "1" ], "equation": { "f": "9*x^6 + 3*x^5 - 39*x^4 + 5*x^3 + 46*x^2 - 25*x", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = 9*x^6 + 3*x^5 - 39*x^4 + 5*x^3 + 46*x^2 - 25*x" }, "conductor": "9900", "conductor_sort": 9900, "conductor_note": "", "discriminant_digits": 15, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": 2026, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "generators", "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)", "new_group": false, "new_for_class": true, "certificate": "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); simple over ℚ: χ_47 = T^4 - 34*T^2 + 2209 is irreducible", "verified": "2026-10-17 06:12:07 UTC", "submitted": "2026-10-16", "source_kind": "file" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": 0, "undecided": null }, "first_source": { "simple": "BookerSutherland", "qsplit": "BSSVY", "gsplit": "BookerSutherland", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the odd M₁(8,2,2) family of the lab inventory, f = ((1−u)x+1)((1−v)x+1)((u+v+2)x+1)(−x² + (u²v−u²+uv²−u−v²−v−2)x − (u²+uv+v²+u+v+1)), Q = (−1, uv(u+v+1)), slice v = 3: [2,2,8] is marked over ℚ(u). Every u ≡ 11 (mod 371) has #J(𝔽₇) = 64 and #J(𝔽₅₃) = 3296, gcd 32, strict at 53 (family_228_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings (Bröker–Howe–Lauter–Stevenhagen normal form) of E = 210e2 (torsion ℤ/2 × ℤ/8) with the partners F that have a rational 2-torsion point. The partners anti-symplectically 3-congruent to E form a rational curve, and so does the cover on which F has a rational 2-torsion point (Γ(3) ∩ Γ₀(2) has genus 0); this gives C_v over ℚ(v) with explicit degree-3 maps onto E and F_v, so J is isogenous over ℚ to E × F_v. The map onto E has prime degree, so pullback is injective on E and [2,8] ⊂ J(ℚ); over ℚ(v) the quadratic divisors of f_v give J(ℚ)[2] of 2-rank 3, one more, so [2,2,8] is marked. Every 11- and 13-integral v ≡ −2 (mod 143) has the reductions of v = −2 at 11 and 13, where J(𝔽₁₁) ≅ [2,2,40] and J(𝔽₁₃) ≅ [2,8,16]; the largest group embedding in both is [2,2,8], so J(ℚ)_tors is exactly [2,2,8]; the G2-invariants at v = −2 and v = 141 differ (family_2_2_8_split_certificate.log)", "inherited_from": null } } }, { "key": "2.2.10", "group": [ 2, 2, 10 ], "bracket": "[2,2,10]", "label": "ℤ/2 ⊕ ℤ/2 ⊕ ℤ/10", "order": 40, "rank": 3, "classes": { "simple": [ { "id": "2.2.10.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "0", "-7", "12", "6", "-12" ], "h": [ "1", "0", "1" ], "equation": { "f": "-12*x^5 + 6*x^4 + 12*x^3 - 7*x^2", "h": "x^2 + 1", "text": "y^2 + (x^2 + 1)*y = -12*x^5 + 6*x^4 + 12*x^3 - 7*x^2" }, "conductor": "10512", "conductor_sort": 10512, "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", "discriminant_digits": 10, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 61 (source: \\cite{BookerSutherland})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 17 is strict (χ = T^4 + 2*T^3 - 6*T^2 + 34*T + 289)", "verified": "2026-09-18 17:32:57 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "2.2.10.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "-20", "-12", "66", "-2", "-45", "3", "9" ], "h": [ "0", "1", "1" ], "equation": { "f": "9*x^6 + 3*x^5 - 45*x^4 - 2*x^3 + 66*x^2 - 12*x - 20", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = 9*x^6 + 3*x^5 - 45*x^4 - 2*x^3 + 66*x^2 - 12*x - 20" }, "conductor": "30492", "conductor_sort": 30492, "conductor_note": "", "discriminant_digits": 16, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 65 (source: \\cite{BookerSutherland})", "new_group": false, "new_for_class": true, "certificate": "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 - 5/6, hh = 324*x^3 + 972*x^2 + 432*x - 432; degree-3 map to the elliptic curve 66c1 over Q [Genus2Torsion/verify_split_certificates.m, cover 16]", "verified": "2026-09-18 17:40:53 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "BookerSutherland", "qsplit": "BookerSutherland", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 2, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite", "Elkies2024" ], "note": "Elkies' 5-torsion chart with four rational Weierstrass points on the nodal cubic a³ + 5a² − (5/9)ab² + (1/3)b² = 0, parametrised by τ: D₅ = [Q, x] has order 5 and the root-pair classes generate (ℤ/2)³, so [2,2,10] is marked over ℚ(τ). Every τ ≡ 57 (mod 65) has #J(𝔽₅) = 40 and #J(𝔽₁₃) = 120 (gcd 40), strict at 13 (family_2210_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings (Bröker–Howe–Lauter–Stevenhagen normal form) of E = 66c1 (torsion ℤ/10) with the partners F that have full rational 2-torsion, parametrised by the genus-1 curve D: s² = −2v⁴ + 19v³ − 12v² − 80v + 196, whose Jacobian y² = x³ + 39204 has the point P = (616, −15290) of infinite order. Explicit degree-3 maps make J isogenous over ℚ to E × F_v; pullback is injective on torsion of order prime to 3, and E and F_v are not isogenous (#E(𝔽₂₃) = 30, #F_v(𝔽₂₃) = 24), so [2,2,10] ⊂ J(ℚ). For R_k = R₀ + 48kP (R₀ over v = 0, the census curve 2.2.10.b, the gluing of 66c1 and 462f2), v(R_k) ≡ 0 modulo 23 and 47, where #J(𝔽₂₃) = 720 and #J(𝔽₄₇) = 2200, so J(ℚ)_tors is exactly [2,2,10]; the G2-invariants at v = 0 and v = 5560/3033 differ (family_2210_split_certificate.log)", "inherited_from": null } } }, { "key": "2.2.12", "group": [ 2, 2, 12 ], "bracket": "[2,2,12]", "label": "ℤ/2 ⊕ ℤ/2 ⊕ ℤ/12", "order": 48, "rank": 3, "classes": { "simple": [ { "id": "2.2.12.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "-5", "-5", "16", "8", "-7", "1" ], "h": [ "0", "1", "1" ], "equation": { "f": "x^6 - 7*x^5 + 8*x^4 + 16*x^3 - 5*x^2 - 5*x", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = x^6 - 7*x^5 + 8*x^4 + 16*x^3 - 5*x^2 - 5*x" }, "conductor": "12300", "conductor_sort": 12300, "conductor_note": "", "discriminant_digits": 14, "lmfdb": { "label": "12300.e.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C-5%2C-5%2C16%2C8%2C-7%2C1%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "12300.e.2" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 62 (source: \\cite{BookerSutherland})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 19 is strict (χ = T^4 + 2*T^3 - 18*T^2 + 38*T + 361)", "verified": "2026-09-18 17:32:58 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "2.2.12.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "1", "-9", "19", "19", "-55", "-45" ], "h": [ "0", "1", "1" ], "equation": { "f": "-45*x^5 - 55*x^4 + 19*x^3 + 19*x^2 - 9*x + 1", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = -45*x^5 - 55*x^4 + 19*x^3 + 19*x^2 - 9*x + 1" }, "conductor": "1350", "conductor_sort": 1350, "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)", "discriminant_digits": 8, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 66 (source: \\cite{BookerSutherland})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:40:54 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "BookerSutherland", "qsplit": "BookerSutherland", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 2, "infinite": { "simple": { "grade": "exact", "sources": [ "ThisSite", "BNSS2026" ], "note": "y² = x(x² − Sx + w²)(x − a²)(x − b²)(x − c²) with S = a² + b² + c², w² = a²b² + a²c² + b²c² (as in BNSS2026, F₁ − F₂ = (abc)², so D₃ = ∞₊ − ∞₋ has order 3), on the conic a = m² − 1, b = 2m + 1, c = 2(m² + m + 1), w = −(2a² + ab + 2b²): since S + 2w = (a + b)² is a square, the class [x² − Sx + w², 0] of the irrational root pair has an explicit half D₄ over ℚ(m), and with [x(x − a²), 0], [x(x − b²), 0] it marks [2,2,12]. Every m ≡ 3 (mod 1357) has #J(𝔽₂₃) = 528 and #J(𝔽₅₉) = 3984 (gcd 48), strict at 59 (family_2_2_12_certificate.log). A second family, over the rank-1 elliptic curve A² = τ⁴ + τ³ − 17τ² + 7τ + 49 (Jacobian of conductor 2660), contains the census curve 2.2.12.a: for R = 36k·P it has the reductions of 2.2.12.a at 17 and 233, #J = 240 and 56496 (gcd 48), strict at 233 (family_2_2_12_control_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the (3,3)-gluings (Bröker–Howe–Lauter–Stevenhagen normal form) of E = 90c3 (torsion ℤ/12) with the partners F that have full rational 2-torsion. They are parametrised by the genus-1 curve D: s² = v⁴ − 4v³ + 6v² − 652v + 649, whose Jacobian y² = x³ + 9 has the point P = (3, 6) of infinite order. Explicit degree-3 maps make J isogenous over ℚ to E × F_v; pullback along the map onto E is injective (prime degree) and along the map onto F_v is injective on F_v(ℚ)[2], and the two images meet only in 3-torsion, so [2,2,12] ⊂ J(ℚ) (F_v inherits a rational point of order 3 from E, but no extra 3-torsion appears in J). For R_k = R₀ + 21kP (R₀ = 2P over v = 101/5), v(R_k) ≡ 101/5 modulo 13 and 83, where J(𝔽₁₃) ≅ [2,6,12] (2-primary part [2,2,4]) and J(𝔽₈₃) ≅ [2,2,2,1008] (3-primary part ℤ/9), so J(ℚ)_tors is exactly [2,2,12]; the G2-invariants at v = 101/5 and at v = 8567/629 (over 3P) differ (family_2_2_12_split_certificate.log)", "inherited_from": null } } }, { "key": "2.2.14", "group": [ 2, 2, 14 ], "bracket": "[2,2,14]", "label": "ℤ/2 ⊕ ℤ/2 ⊕ ℤ/14", "order": 56, "rank": 3, "classes": { "simple": [ { "id": "2.2.14.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "-150965100", "-169058940", "-16976459", "1218410", "96973", "-4020", "36" ], "h": [ "0" ], "equation": { "f": "36*x^6 - 4020*x^5 + 96973*x^4 + 1218410*x^3 - 16976459*x^2 - 169058940*x - 150965100", "h": "0", "text": "y^2 = 36*x^6 - 4020*x^5 + 96973*x^4 + 1218410*x^3 - 16976459*x^2 - 169058940*x - 150965100" }, "conductor": "38200383011700", "conductor_sort": 38200383011700, "conductor_note": "", "discriminant_digits": 60, "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": "" }, "sources": [ "BookerSutherland", "BNSS2026" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database); Balakrishnan–Najman–Shnidman–Sutherland 2026", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 63 (source: \\cite{BookerSutherland}, new)", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 17 is strict (χ = T^4 - 4*T^3 + 6*T^2 - 68*T + 289)", "verified": "2026-09-18 17:32:59 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "BookerSutherland", "qsplit": null, "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": false, "gsplit": false, "undecided": false }, "exclusions": { "qsplit": { "reason": "A Jacobian split over ℚ contains an elliptic curve over ℚ with an elliptic quotient over ℚ. By Mazur's theorem, a rational point of order 7 then forces rational 2-rank at most 2, and rational 2-rank 4 excludes rational points of orders 5 and 7.", "sources": [ "QSplitTwoRankObstruction", "Mazur1977" ], "proof_url": "proofs/qsplit-two-rank-obstruction.html" }, "gsplit": { "reason": "If a geometrically split abelian surface over ℚ is simple over ℚ and has a rational point of order 7, its rational 2-torsion has rank at most 2. The proof applies to every geometric splitting degree and requires simplicity over ℚ.", "sources": [ "QSimple2214Obstruction", "QSimple63Obstruction", "KenkuMomose1988", "SerreTate1968" ], "proof_url": "proofs/qsimple-2214-obstruction.html" } }, "n_curves": 1, "infinite": { "simple": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the contact-7 chart with three rational roots 1 − s₁², 1 − u², 1 − s₃²: at s₁ = −1/2 the pairs (u, s₃) form a genus-1 curve birational to E: y² = x³ − 12x (rank 1, P = (−2, 4)), and [2,2,14] is marked over its function field. For R = (1 + 26k)P (26 = lcm of the orders of P mod 17 and mod 29) the curve has the reduction of R = P, (u, s₃) = (−13/49, 13/50), at 17 and 29: #J(𝔽₁₇) = 392, #J(𝔽₂₉) = 1120, gcd 56, strict at 17 (family_2214_certificate.log)", "inherited_from": null }, "split": { "grade": "impossible", "sources": [ "QSimple2214Obstruction", "QSimple63Obstruction", "KenkuMomose1988", "SerreTate1968", "QSplitTwoRankObstruction", "Mazur1977" ], "note": "A Jacobian split over ℚ contains an elliptic curve over ℚ with an elliptic quotient over ℚ. By Mazur's theorem, a rational point of order 7 then forces rational 2-rank at most 2, and rational 2-rank 4 excludes rational points of orders 5 and 7. If a geometrically split abelian surface over ℚ is simple over ℚ and has a rational point of order 7, its rational 2-torsion has rank at most 2. The proof applies to every geometric splitting degree and requires simplicity over ℚ.", "proof_url": "proofs/qsimple-2214-obstruction.html", "inherited_from": null } } }, { "key": "2.2.16", "group": [ 2, 2, 16 ], "bracket": "[2,2,16]", "label": "ℤ/2 ⊕ ℤ/2 ⊕ ℤ/16", "order": 64, "rank": 3, "classes": { "simple": [ { "id": "2.2.16.b", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "-284225007", "519681459", "166739244", "-496170642", "128899980", "-11736600", "360000" ], "h": [ "0", "1", "1" ], "equation": { "f": "360000*x^6 - 11736600*x^5 + 128899980*x^4 - 496170642*x^3 + 166739244*x^2 + 519681459*x - 284225007", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = 360000*x^6 - 11736600*x^5 + 128899980*x^4 - 496170642*x^3 + 166739244*x^2 + 519681459*x - 284225007" }, "conductor": "647548862090857026300", "conductor_sort": 647548862090857026300, "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)", "discriminant_digits": 82, "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": "" }, "sources": [], "rm": false, "discovered_by": "A. V. Sutherland", "credit_differs": false, "year": 2026, "historical": false, "first_source": null, "first_source_year": null, "torsion_method": "TorsionSubgroup", "reference": "A. V. Sutherland, personal communication to F. Najman (November 2026)", "new_group": false, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 47 is strict (χ = T^4 - 6*T^3 - 2*T^2 - 282*T + 2209)", "verified": "2026-11-10 08:49:11 UTC", "submitted": "2026-11-10", "source_kind": "file" } ], "qsplit": [ { "id": "2.2.16.a", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "9", "-33", "3", "61", "3", "-33", "-11" ], "h": [ "0", "1", "1" ], "equation": { "f": "-11*x^6 - 33*x^5 + 3*x^4 + 61*x^3 + 3*x^2 - 33*x + 9", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = -11*x^6 - 33*x^5 + 3*x^4 + 61*x^3 + 3*x^2 - 33*x + 9" }, "conductor": "69300", "conductor_sort": 69300, "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)", "discriminant_digits": 17, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 67 (source: \\cite{BookerSutherland})", "new_group": true, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:40:56 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": 0, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": null, "qsplit": "BookerSutherland", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 2, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4) of E₁ = E_{2,4}(t) with E₂ = 210e2 (torsion ℤ/2 × ℤ/8) along the identity on the labelled 2-torsion (special points matched), C_t: y² = −(28t − 25)(100t − 7)(324x² − (4t − 1)²)(1024x² − (4t + 1)²)(175x² − 4t), with (x, y) ↦ (x², y) onto E₂ and (x, y) ↦ (1/x², y/x³) onto E₁(t), so J is isogenous over ℚ to E₁ × E₂; [2,2,8] is marked over ℚ(t). On the conic t + 1/4 = 5c², t − 1/4 = 30e², parametrised by c = (12n² − 6n + 2)/(30n² − 5), the Kummer class of the point of order 4 of E₁ equals that of P + (−36, 18), P = (244, 3658) of order 8 on E₂ (the [2,2,2,16] construction without its third condition), and the sum of the pullbacks of their halves, computed over ℚ(√10, √−15)(n), has coefficients in ℚ(n) and order 16, so [2,2,16] is marked over ℚ(n). Here #J(𝔽_p) is always divisible by 128, so the group structure is used instead: every 13- and 19-integral n ≡ 83 (mod 247) has J(𝔽₁₃) ≅ [2,8,16] and J(𝔽₁₉) ≅ [2,2,2,48], so J(ℚ)_tors is exactly [2,2,16]; the G2-invariants of the members differ. The census curve 2.2.16.a is the member n = 2/3 (t = 11/20), in the class (family_2_2_16_split_certificate.log)", "inherited_from": null } } }, { "key": "2.2.20", "group": [ 2, 2, 20 ], "bracket": "[2,2,20]", "label": "ℤ/2 ⊕ ℤ/2 ⊕ ℤ/20", "order": 80, "rank": 3, "classes": { "simple": [ { "id": "2.2.20.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "1016064", "10501344", "15616273", "-60534098", "27387697", "7579404", "-1566684" ], "h": [ "0" ], "equation": { "f": "-1566684*x^6 + 7579404*x^5 + 27387697*x^4 - 60534098*x^3 + 15616273*x^2 + 10501344*x + 1016064", "h": "0", "text": "y^2 = -1566684*x^6 + 7579404*x^5 + 27387697*x^4 - 60534098*x^3 + 15616273*x^2 + 10501344*x + 1016064" }, "conductor": "16897247339672700", "conductor_sort": 16897247339672700, "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", "discriminant_digits": 74, "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": "" }, "sources": [ "BNSS2026" ], "rm": false, "discovered_by": "Balakrishnan–Najman–Shnidman–Sutherland 2026", "credit_differs": false, "year": null, "historical": false, "first_source": "BNSS2026", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 64 (source: new)", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 71 is strict (χ = T^4 + 2*T^3 + 14*T^2 + 142*T + 5041)", "verified": "2026-09-18 17:33:00 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "BNSS2026", "qsplit": null, "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": false, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [ "BNSS2026", "Najman2026" ], "note": "the displayed curve is the unique one in its one-parameter family (BNSS2026, §'A second order-80 group', by Chabauty); Elkies' family gives infinitely many Jacobians of order 80, but with group [2,2,2,10]", "inherited_from": null }, "split": { "grade": "open", "sources": [], "note": "", "inherited_from": null } } }, { "key": "2.2.24", "group": [ 2, 2, 24 ], "bracket": "[2,2,24]", "label": "ℤ/2 ⊕ ℤ/2 ⊕ ℤ/24", "order": 96, "rank": 3, "classes": { "simple": [], "qsplit": [ { "id": "2.2.24.a", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "0", "-158457693276225", "-25400003349776", "19670508870560", "-10404361990400", "4237596000000" ], "h": [ "0" ], "equation": { "f": "4237596000000*x^5 - 10404361990400*x^4 + 19670508870560*x^3 - 25400003349776*x^2 - 158457693276225*x", "h": "0", "text": "y^2 = 4237596000000*x^5 - 10404361990400*x^4 + 19670508870560*x^3 - 25400003349776*x^2 - 158457693276225*x" }, "conductor": "206076647700", "conductor_sort": 206076647700, "conductor_note": "N(E1) N(E2) = 825330 * 249690 for the elliptic curves ['1', '0', '0', '-1362433018010', '-603188297771007900'] and ['1', '0', '0', '-673039570', '-5895575919100'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "discriminant_digits": 117, "lmfdb": { "label": "", "kind": "jump", "url": "https://www.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C-158457693276225%2C-25400003349776%2C19670508870560%2C-10404361990400%2C4237596000000%5D%2C%5B0%5D%5D", "snapshot_label": "" }, "sources": [ "HLP2000" ], "rm": false, "discovered_by": "Balakrishnan–Najman–Shnidman–Sutherland 2026", "credit_differs": true, "year": null, "historical": false, "first_source": "HLP2000", "first_source_year": 2000, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 68 (source: \\cite{HLP2000})", "new_group": true, "new_for_class": true, "certificate": "an involution over Q other than the hyperelliptic one, (x : y : z) -> (x + 3969/2500*z : 1826585575424/962966796875*y : 52316/156025*x - z) on the model y^2 + h y = f; its quotient is a genus-1 curve over Q", "verified": "2026-09-18 17:40:58 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": null, "qsplit": "HLP2000", "gsplit": null, "undecided": null }, "known": { "simple": false, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite", "HLP2000" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4, §3.7) of E = E_{2,6}(241/81) (conductor 825330) with the Kubert curves F_u = E_{2,8}(u) along ψ(T₁) = T₂′, ψ(T₂) = T₃′, ψ(T₃) = T₁′ (HLP Table 5), C_u: y² = c₂F(kx² − 1, c₂x²) over ℚ(u) (F(X, Z) = Z³f(X/Z), φ the quadratic with φ(x(T_i)) = x(T′_ψ(i))). The maps (x, y) ↦ (1/x², y/x³), (x², y) onto E and F_u make J isogenous over ℚ to E × F_u, and [2,24] is marked over ℚ(u). On HLP's gain curve D: s² = −41(8u² − 1)(8u² + 8u + 1) (Jacobian y² = x³ − 1681x, rank 2) two roots U₂, U₃ of the cubic in x² become squares, giving a new rational 2-torsion class: [2,2,24]. For R = R₀ + 120kP (u(R₀) = 1/3, P = (−9, −120)) u(R) ≡ 1/3 modulo 13 and 47, where J(𝔽₁₃) = [2,2,48] and J(𝔽₄₇) = [2,2,24,24], so J(ℚ)_tors is exactly [2,2,24]; the G2-invariants at u = 1/3 and u = 5/27 differ. The member u = 1/3 is the census curve 2.2.24.a (family_2_2_24_split_certificate.log)", "inherited_from": null } } }, { "key": "2.4.4", "group": [ 2, 4, 4 ], "bracket": "[2,4,4]", "label": "ℤ/2 ⊕ ℤ/4 ⊕ ℤ/4", "order": 32, "rank": 3, "classes": { "simple": [ { "id": "2.4.4.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "180", "-210", "-1850", "2204", "-116" ], "h": [ "0", "1", "1" ], "equation": { "f": "-116*x^5 + 2204*x^4 - 1850*x^3 - 210*x^2 + 180*x", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = -116*x^5 + 2204*x^4 - 1850*x^3 - 210*x^2 + 180*x" }, "conductor": "172260", "conductor_sort": 172260, "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", "discriminant_digits": 29, "lmfdb": { "label": "172260.bj.3", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C180%2C-210%2C-1850%2C2204%2C-116%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "172260.bj.3" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 65 (source: \\cite{BookerSutherland})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 37 is strict (χ = T^4 - 4*T^3 - 2*T^2 - 148*T + 1369)", "verified": "2026-09-18 17:33:01 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "2.4.4.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "0", "15", "40", "0", "161", "-240" ], "h": [ "0", "1" ], "equation": { "f": "-240*x^5 + 161*x^4 + 40*x^2 + 15*x", "h": "x", "text": "y^2 + x*y = -240*x^5 + 161*x^4 + 40*x^2 + 15*x" }, "conductor": "225", "conductor_sort": 225, "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", "discriminant_digits": 21, "lmfdb": { "label": "225.a.5", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C15%2C40%2C0%2C161%2C-240%5D%2C%5B0%2C1%5D%5D", "snapshot_label": "225.a.5" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 69 (source: \\cite{BookerSutherland})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:40:59 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "BookerSutherland", "qsplit": "BookerSutherland", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 2, "infinite": { "simple": { "grade": "exact", "sources": [ "ThisSite", "Nicholls2018", "Genus2TorsionNotebook" ], "note": "Nicholls' family Λ₃₃₄ (as built in the Genus2TorsionNotebook), on its rational locus where l₃ = x² − a splits (a = w², w = (m²t² − t⁴ + t²)/((m² + t² − 1)t), s = m(w − t)), slice t = m = 2 with parameter v: two order-4 classes from halving identities F = cu² + Rl² for R = l₃ and R = l₂, so [2,4,4] is marked over ℚ(v). Every v ≡ 2 (mod 589) has the reductions of v = 2 at 19 and 31: #J(𝔽₁₉) = 288, #J(𝔽₃₁) = 1088, gcd 32 = #[2,4,4], strict at 31 (family_244_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "a Howe–Leprévost–Poonen (2,2)-self-gluing (HLP 2000, Prop. 4; the construction of their Corollary 6) of a curve with ℤ/2 × ℤ/4: C_s: y² = −(s⁴ + 1)(x² + 1)(x² − s⁴)(s⁴x² − 1) = F(x²), F(X) = X³F(1/X), so (x, y) ↦ (x², y) and (x, y) ↦ (1/x², y/x³) both map C_s onto E_s: Y² = F(X), and J is isogenous over ℚ to E_s × E_s. E_s is the Kubert curve E_{2,4}(t) at t = m/(2(m² + 1)), m = (s + 1)/(s − 1), and the gluing fixes 2P and swaps the other two points of order 2. On this locus the Kummer class of the point P of order 4 is invariant under the swap (1 − 16t² is a square), so the halves of P are defined over a quadratic field and the pullback of a half to both factors is a rational class of order 4; with the pullback of P (order 4) and [x² − s⁴, 0], [2,4,4] is marked over ℚ(s). Here #J(𝔽_p) = #E_s(𝔽_p)² is always divisible by 64, so the group structure is used instead: every s ≡ 86 (mod 287), integral at 7 and 41, has J(𝔽₇) ≅ [2,4,8] and J(𝔽₄₁) ≅ [4,4,12,12], and the largest group embedding in both is [2,4,4], so J(ℚ)_tors is exactly [2,4,4]. The census curve 2.4.4.b is the member s = 2; the G2-invariants of the members differ (family_2_4_4_split_certificate.log)", "inherited_from": null } } }, { "key": "2.4.8", "group": [ 2, 4, 8 ], "bracket": "[2,4,8]", "label": "ℤ/2 ⊕ ℤ/4 ⊕ ℤ/8", "order": 64, "rank": 3, "classes": { "simple": [ { "id": "2.4.8.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "-959040", "522729", "4402", "15929", "240" ], "h": [ "0" ], "equation": { "f": "240*x^5 + 15929*x^4 + 4402*x^3 + 522729*x^2 - 959040*x", "h": "0", "text": "y^2 = 240*x^5 + 15929*x^4 + 4402*x^3 + 522729*x^2 - 959040*x" }, "conductor": "10033323300", "conductor_sort": 10033323300, "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", "discriminant_digits": 48, "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": "" }, "sources": [ "BNSS2026" ], "rm": false, "discovered_by": "Balakrishnan–Najman–Shnidman–Sutherland 2026", "credit_differs": false, "year": null, "historical": false, "first_source": "BNSS2026", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 66 (source: new)", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 47 is strict (χ = T^4 - 4*T^3 + 30*T^2 - 188*T + 2209)", "verified": "2026-09-18 17:33:02 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "2.4.8.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "7524", "-1530", "-10562", "797", "3630", "140" ], "h": [ "0", "1", "1" ], "equation": { "f": "140*x^5 + 3630*x^4 + 797*x^3 - 10562*x^2 - 1530*x + 7524", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = 140*x^5 + 3630*x^4 + 797*x^3 - 10562*x^2 - 1530*x + 7524" }, "conductor": "573300", "conductor_sort": 573300, "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)", "discriminant_digits": 34, "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" }, "sources": [ "HLP2000" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "HLP2000", "first_source_year": 2000, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 70 (source: \\cite{HLP2000})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:41:01 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "BNSS2026", "qsplit": "HLP2000", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 2, "infinite": { "simple": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the M₁(8,2^w) odd model y² = xA(x)B(x) of the [4,8] tangent cover (A = x² + (R³ + 4R²t + R − 8Rt + 4t)x + R⁴, B = (R + 2 + 4t)x² + (R² + 4R + 1 + 8t)x + 2R² + R + 4t, t = (2R² + (1 − w²)R − 2w²)/(4(w² − 1))), in which P₈ = (−R, y₀) − ∞ has order 8, restricted to the line w = (13R − 12)/(12R − 13), R = 3(τ² − 4)/(2(τ² − 9)). There both 2-descent conditions for halving T₀ = [x, 0] hold identically, so the tangent half H (2H = T₀) is defined over ℚ(τ) and ⟨P₈, H⟩ = [4,8]. B splits over the genus-1 curve D: σ² = τ⁴ − 36τ² + 36, whose Jacobian y² = x³ − 36x (conductor 576) has rank 1, and [x − β, 0] for a root β of B completes [2,4,8] over ℚ(D). For R_k = R₀ + 60kP (R₀ over τ = 12/35, P from τ = −1), τ(R_k) ≡ 12/35 modulo 11 and 53, where J(𝔽₁₁) ≅ [2,4,16] and J(𝔽₅₃) ≅ [2,4,328] (#J = 128 and 2624, gcd 64), strict at 53, so J(ℚ)_tors is exactly [2,4,8]; the G2-invariants at R₀ and at 60P differ (family_2_4_8_certificate.log). The census curve 2.4.8.a, (R, w) = (−16/11, 14/11), where A splits instead of B, is not on D", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite", "HLP2000" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4) of E₁ = 210e2 (torsion ℤ/2 × ℤ/8) with the Kubert curve E₂ = E_{2,8}(t), along a bijection of their rational 2-torsion matching the points 4P₈ (the other bijections give [8,8]). Up to scaling the HLP sextic is C_t: y² = F(x²), with (x, y) ↦ (x², y) onto E₂ and (x, y) ↦ (1/x², y/x³) onto E₁, so J is isogenous over ℚ to E₁ × E₂. Over ℚ(t) the pullbacks of the two points of order 8 (glued along 4P₈) give [4,8], and the pullback of another 2-torsion point of E₁ completes [2,4,8]. Here #J(𝔽_p) is always divisible by 256, so the group structure is used: every 23- and 41-integral t ≡ 3 (mod 943) has J(𝔽₂₃) ≅ [2,2,4,32] and J(𝔽₄₁) ≅ [8,8,24] (2-exponent 8), so J(ℚ)_tors is exactly [2,4,8]; the G2-invariants of the members differ (family_2_4_8_split_certificate.log)", "inherited_from": null } } }, { "key": "2.6.6", "group": [ 2, 6, 6 ], "bracket": "[2,6,6]", "label": "ℤ/2 ⊕ ℤ/6 ⊕ ℤ/6", "order": 72, "rank": 3, "classes": { "simple": [], "qsplit": [ { "id": "2.6.6.a", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "18", "-54", "9", "71", "-40", "-5", "25" ], "h": [ "0", "1", "1" ], "equation": { "f": "25*x^6 - 5*x^5 - 40*x^4 + 71*x^3 + 9*x^2 - 54*x + 18", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = 25*x^6 - 5*x^5 - 40*x^4 + 71*x^3 + 9*x^2 - 54*x + 18" }, "conductor": "132300", "conductor_sort": 132300, "conductor_note": "N(E1) N(E2) = 210 * 630 for the elliptic curves ['1', '0', '1', '-578', '2756'] and ['1', '-1', '0', '-24309', '1426113'] to whose product J is isogenous over Q (Magma's genus-2 Conductor agrees)", "discriminant_digits": 23, "lmfdb": { "label": "132300.eo.2", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B18%2C-54%2C9%2C71%2C-40%2C-5%2C25%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "132300.eo.2" }, "sources": [ "HLP2000" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": true, "year": null, "historical": false, "first_source": "HLP2000", "first_source_year": 2000, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 71 (source: \\cite{HLP2000})", "new_group": true, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:41:02 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": null, "qsplit": "HLP2000", "gsplit": null, "undecided": null }, "known": { "simple": false, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite", "HLP2000" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4 and §3.7) of E₁ = E_{2,6}(t) with E₂ = E_{2,6}(u) along the identity on the 2-torsion points labelled as in HLP Table 5, with t = 3 + (u − 3)y² and (y, z) the section 2σ of HLP's system (6), ((u − 3)y² + 6)((u − 3)y² − 2) = (u + 3)(u − 5)z², origin (1, 1), σ = (−1, 1): y = (−u⁴ + 12u³ − 126u² + 348u + 279)/(3u⁴ − 20u³ − 54u² + 540u − 981). Then (t − 3)(u − 3) and (t + 3)(t − 5)(u + 3)(u − 5) are squares, so a half of (T₁, T₁′) gives a new rational point of order 2 (HLP Prop. 12). Up to a square C_u: y² = F(x²), with (x, y) ↦ (x², y) onto E₂ and (x, y) ↦ (1/x², y/x³) onto E₁, so J is isogenous over ℚ to E₁ × E₂. Over ℚ(u) the sextic has four rational roots ((ℤ/2)³) and the pullbacks of the points of order 3 give (ℤ/3)², so [2,6,6] is marked. Here #J(𝔽_p) is always divisible by 144, so the group structure is used: every 11- and 17-integral u ≡ 4 (mod 187) has the reductions of u = 4 at 11 and 17, where J(𝔽₁₁) ≅ [2,2,6,6] and J(𝔽₁₇) ≅ [2,6,24], so J(ℚ)_tors is exactly [2,6,6]; the G2-invariants of the members differ (family_2_6_6_split_certificate.log)", "inherited_from": null } } }, { "key": "2.2.2.2", "group": [ 2, 2, 2, 2 ], "bracket": "[2,2,2,2]", "label": "ℤ/2 ⊕ ℤ/2 ⊕ ℤ/2 ⊕ ℤ/2", "order": 16, "rank": 4, "classes": { "simple": [ { "id": "2.2.2.2.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "-1", "-4", "8", "23", "-27" ], "h": [ "0", "1", "1" ], "equation": { "f": "-27*x^5 + 23*x^4 + 8*x^3 - 4*x^2 - x", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = -27*x^5 + 23*x^4 + 8*x^3 - 4*x^2 - x" }, "conductor": "1575", "conductor_sort": 1575, "conductor_note": "", "discriminant_digits": 11, "lmfdb": { "label": "1575.c.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B0%2C-1%2C-4%2C8%2C23%2C-27%5D%2C%5B0%2C1%2C1%5D%5D", "snapshot_label": "1575.c.1" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 67 (source: \\cite{BookerSutherland})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 31 is strict (χ = T^4 + 4*T^3 - 2*T^2 + 124*T + 961)", "verified": "2026-09-18 17:33:02 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "2.2.2.2.c", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "-196", "0", "249", "0", "-54", "0", "1" ], "h": [ "0" ], "equation": { "f": "x^6 - 54*x^4 + 249*x^2 - 196", "h": "0", "text": "y^2 = x^6 - 54*x^4 + 249*x^2 - 196" }, "conductor": "28350", "conductor_sort": 28350, "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)", "discriminant_digits": 26, "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": "" }, "sources": [], "rm": false, "discovered_by": "audit of 2026-09-18", "credit_differs": false, "year": 2026, "historical": false, "first_source": null, "first_source_year": null, "torsion_method": "TorsionSubgroup", "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", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:41:21 UTC", "submitted": "2026-09-18", "source_kind": "file" } ], "gsplit": [ { "id": "2.2.2.2.d", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "0", "-3", "1", "17", "16", "3" ], "h": [ "0", "1", "1" ], "equation": { "f": "3*x^5 + 16*x^4 + 17*x^3 + x^2 - 3*x", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = 3*x^5 + 16*x^4 + 17*x^3 + x^2 - 3*x" }, "conductor": "1521", "conductor_sort": 1521, "conductor_note": "", "discriminant_digits": 12, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": 2026, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "generators", "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)", "new_group": false, "new_for_class": true, "certificate": "(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); simple over ℚ: χ_11 = T^4 - 10*T^2 + 121 is irreducible", "verified": "2026-10-17 06:01:54 UTC", "submitted": "2026-10-16", "source_kind": "file" } ], "undecided": [ { "id": "2.2.2.2.b", "class": "split_undecided_over_Q", "class_certified": false, "status": "verified", "class_text": "geometrically split; whether it splits over ℚ is undecided", "f": [ "26", "41", "-44", "-30", "25", "-4" ], "h": [ "1", "1" ], "equation": { "f": "-4*x^5 + 25*x^4 - 30*x^3 - 44*x^2 + 41*x + 26", "h": "x + 1", "text": "y^2 + (x + 1)*y = -4*x^5 + 25*x^4 - 30*x^3 - 44*x^2 + 41*x + 26" }, "conductor": "784", "conductor_sort": 784, "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", "discriminant_digits": 16, "lmfdb": { "label": "784.c.1", "kind": "alpha-snapshot", "url": "https://alpha.lmfdb.org/Genus2Curve/Q/?jump=%5B%5B26%2C41%2C-44%2C-30%2C25%2C-4%5D%2C%5B1%2C1%5D%5D", "snapshot_label": "784.c.1" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 72 (source: \\cite{BookerSutherland})", "new_group": false, "new_for_class": false, "certificate": "(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)", "verified": "2026-09-18 17:41:11 UTC", "submitted": "2026-08-30", "source_kind": "import" } ] }, "first_dated": { "simple": null, "qsplit": 0, "gsplit": 0, "undecided": null }, "first_source": { "simple": "BookerSutherland", "qsplit": null, "gsplit": "BookerSutherland", "undecided": "BookerSutherland" }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": true }, "exclusions": {}, "n_curves": 4, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite" ], "note": "y² = x(x−1)(x−2)(x−3)(x−t): J[2] ≅ [2,2,2,2] is rational over ℚ(t). Every t ≡ 5 (mod 77) has #J(𝔽₇) = 48 and #J(𝔽₁₁) = 80 (gcd 16), strict at 11 (family_2222_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "bielliptic curves with six rational Weierstrass points and one elliptic quotient fixed (the Howe–Leprévost–Poonen (2,2)-gluing, HLP 2000, of E = 45a2 and E₂(m) along (eᵢ, 0) ↦ (1/rᵢ², 0), with full gain): C_m: y² = −3m(m − 1)(m − 16)(x² − r₁²)(x² − r₂²)(x² − r₃²), r₁ = m² − 16, r₂ = m² − 2m + 16, r₃ = −m² + 32m − 16 (the lines through (1 : 1 : 1) on the conic r₃² − r₁² = 16(r₂² − r₁²), which makes the quotient by x ↦ −x isomorphic to E: Y² = (u + 4)(u + 13/4)(u − 8)); (x, y) ↦ (μx² + k, μy/s) maps C_m onto E (checked identity) and (x, y) ↦ (1/x², y/x³) onto E₂(m), so J is isogenous over ℚ to E × E₂(m). Over ℚ(m) the six Weierstrass points ±rᵢ are rational, so J[2] is rational and [2,2,2,2] is marked. Every 11- and 17-integral m ≡ −2 (mod 187) has the reductions of m = −2 (the census curve 2.2.2.2.c) at 11 and 17, where J(𝔽₁₁) ≅ [2,2,2,16] and J(𝔽₁₇) ≅ [2,2,10,10]; the largest group embedding in both is [2,2,2,2], so J(ℚ)_tors is exactly [2,2,2,2]; the G2-invariants of the members differ (family_2_2_2_2_split_certificate.log)", "inherited_from": null } } }, { "key": "2.2.2.4", "group": [ 2, 2, 2, 4 ], "bracket": "[2,2,2,4]", "label": "ℤ/2 ⊕ ℤ/2 ⊕ ℤ/2 ⊕ ℤ/4", "order": 32, "rank": 4, "classes": { "simple": [ { "id": "2.2.2.4.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "1", "-6", "-16", "146", "-225" ], "h": [ "0", "1", "1" ], "equation": { "f": "-225*x^5 + 146*x^4 - 16*x^3 - 6*x^2 + x", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = -225*x^5 + 146*x^4 - 16*x^3 - 6*x^2 + x" }, "conductor": "9450", "conductor_sort": 9450, "conductor_note": "", "discriminant_digits": 9, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 68 (source: \\cite{BookerSutherland})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 17 is strict (χ = T^4 - 4*T^3 + 6*T^2 - 68*T + 289)", "verified": "2026-09-18 17:33:03 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "2.2.2.4.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "0", "-21", "-6", "34", "16", "-3" ], "h": [ "0", "1", "1" ], "equation": { "f": "-3*x^5 + 16*x^4 + 34*x^3 - 6*x^2 - 21*x", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = -3*x^5 + 16*x^4 + 34*x^3 - 6*x^2 - 21*x" }, "conductor": "882", "conductor_sort": 882, "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)", "discriminant_digits": 16, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 73 (source: \\cite{BookerSutherland})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:41:12 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [ { "id": "2.2.2.4.c", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "0", "75", "-490", "659", "392", "48" ], "h": [ "0" ], "equation": { "f": "48*x^5 + 392*x^4 + 659*x^3 - 490*x^2 + 75*x", "h": "0", "text": "y^2 = 48*x^5 + 392*x^4 + 659*x^3 - 490*x^2 + 75*x" }, "conductor": "176400", "conductor_sort": 176400, "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", "discriminant_digits": 23, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": 2026, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "generators", "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)", "new_group": false, "new_for_class": true, "certificate": "(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); simple over ℚ: χ_19 = T^4 - 10*T^2 + 361 is irreducible", "verified": "2026-10-17 06:01:56 UTC", "submitted": "2026-10-16", "source_kind": "file" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": 0, "undecided": null }, "first_source": { "simple": "BookerSutherland", "qsplit": "BookerSutherland", "gsplit": "BookerSutherland", "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite", "BNSS2026" ], "note": "y² = x(x+1)(x+4)(x+9)(x+t²) (the square-branch chart of BNSS2026, Lemma 2.3): a half D₀ of [x, 0] and the classes [x+1, 0], [x+4, 0], [x+9, 0] mark [2,2,2,4] over ℚ(t). Every t ≡ 5 (mod 649) has #J(𝔽₁₁) = 128 and #J(𝔽₅₉) = 4000 (gcd 32), strict at 59 (family_2224_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "bielliptic curves with six rational Weierstrass points and the quotient E = 21a1 (torsion ℤ/2 × ℤ/4) fixed (the Howe–Leprévost–Poonen (2,2)-gluing, HLP 2000, of E and E₂(m), with full gain): C_m: y² = −m(m − 1)(7m − 16)(x² − r₁²)(x² − r₂²)(x² − r₃²), r₁ = m² − 16/7, r₂ = m² − 2m + 16/7, r₃ = −m² + 32m/7 − 16/7 (the lines through (1 : 1 : 1) on the conic r₃² − r₁² = (16/7)(r₂² − r₁²)); (x, y) ↦ (μx² + k, μy/s) maps C_m onto E: Y² = (u + 2)(u + 1/4)(u − 2) and (x, y) ↦ (1/x², y/x³) onto E₂(m), so J is isogenous over ℚ to E × E₂(m). Over ℚ(m), J[2] is rational and the pullback of the point (−1, −1) of order 4 has order 4, so [2,2,2,4] is marked. E₂(m) has the same 2-Kummer images as E, so it also contains ℤ/2 × ℤ/4 and #J(𝔽_p) is always divisible by 64; the group structure is used instead: every 11- and 101-integral m ≡ −2 (mod 1111) has the reductions of m = −2 (the census curve 2.2.2.4.b, with E₂ = 42a2) at 11 and 101, where J(𝔽₁₁) ≅ [2,2,2,16] and J(𝔽₁₀₁) ≅ [2,2,4,572]; the largest group embedding in both is [2,2,2,4], so J(ℚ)_tors is exactly [2,2,2,4]; the G2-invariants of the members differ. The member m = 2 (outside the class) is the census curve 2.2.2.8.b (family_2_2_2_4_split_certificate.log)", "inherited_from": null } } }, { "key": "2.2.2.6", "group": [ 2, 2, 2, 6 ], "bracket": "[2,2,2,6]", "label": "ℤ/2 ⊕ ℤ/2 ⊕ ℤ/2 ⊕ ℤ/6", "order": 48, "rank": 4, "classes": { "simple": [ { "id": "2.2.2.6.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "2", "-12", "11", "18", "-14", "-6" ], "h": [ "1", "0", "1" ], "equation": { "f": "-6*x^5 - 14*x^4 + 18*x^3 + 11*x^2 - 12*x + 2", "h": "x^2 + 1", "text": "y^2 + (x^2 + 1)*y = -6*x^5 - 14*x^4 + 18*x^3 + 11*x^2 - 12*x + 2" }, "conductor": "39600", "conductor_sort": 39600, "conductor_note": "", "discriminant_digits": 13, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 69 (source: \\cite{BookerSutherland})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 23 is strict (χ = T^4 - 4*T^3 - 2*T^2 - 92*T + 529)", "verified": "2026-09-18 17:33:04 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "2.2.2.6.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "0", "30", "39", "-72", "-111", "-30" ], "h": [ "0", "1", "1" ], "equation": { "f": "-30*x^5 - 111*x^4 - 72*x^3 + 39*x^2 + 30*x", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = -30*x^5 - 111*x^4 - 72*x^3 + 39*x^2 + 30*x" }, "conductor": "900", "conductor_sort": 900, "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)", "discriminant_digits": 19, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 74 (source: \\cite{BookerSutherland})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:41:14 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [ { "id": "2.2.2.6.c", "class": "gsplit", "class_certified": true, "status": "certified", "class_text": "geometrically split, simple over ℚ", "f": [ "1355999975081200", "2757085929534360", "16095094765893", "14380189111", "-25655829", "6993" ], "h": [ "0", "1", "1" ], "equation": { "f": "6993*x^5 - 25655829*x^4 + 14380189111*x^3 + 16095094765893*x^2 + 2757085929534360*x + 1355999975081200", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = 6993*x^5 - 25655829*x^4 + 14380189111*x^3 + 16095094765893*x^2 + 2757085929534360*x + 1355999975081200" }, "conductor": "168017780700", "conductor_sort": 168017780700, "conductor_note": "", "discriminant_digits": 95, "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": "" }, "sources": [], "rm": false, "discovered_by": "F. Najman", "credit_differs": false, "year": 2026, "historical": false, "first_source": null, "first_source_year": null, "torsion_method": "generators", "reference": "new", "new_group": false, "new_for_class": true, "certificate": "(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); simple over ℚ: χ_29 = T^4 - 26*T^2 + 841 is irreducible", "verified": "2026-10-17 06:01:57 UTC", "submitted": "2026-10-16", "source_kind": "file" } ], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": 0, "undecided": null }, "first_source": { "simple": "BookerSutherland", "qsplit": "BookerSutherland", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": true, "undecided": false }, "exclusions": {}, "n_curves": 3, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite" ], "note": "the M(2,2,2,6) chart: y² = x(x+2s²−sn)(x+2s²+sm−2sn−mn)(x+2s²+sm−sn−mn)(2x−mn)(2x+4s²−4sn−mn), on the line s = 7, m = 2: a class of order 6 and the rational J[2] mark [2,2,2,6] over ℚ(n). Every n ≡ 278 (mod 299) has #J(𝔽₁₃) = 240 and #J(𝔽₂₃) = 432 (gcd 48), strict at 23 (family_2226_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4) of the Kubert curve E₂ = E_{2,6}(s + 3) with its 3-isogenous curve E₁ = E₂/⟨2P⟩, along the isomorphism of 2-torsion induced by the dual isogeny (the Kummer maps commute with the isogeny, so the gain is full): C_s: y² = (s² + 12)(s² − 4x²)((s − 2)²x² − (s + 6)²)((s + 2)²x² − (s − 6)²), with six rational Weierstrass points and maps (x, y) ↦ (x², y) onto E₂ and (x, y) ↦ (1/x², y/x³) onto E₁, so J is isogenous over ℚ to E₁ × E₂. Over ℚ(s), J[2] is rational and the pullback of the point of order 3 of E₂ has order 3, so [2,2,2,6] is marked. Every 13-integral s ≡ 4 (mod 13) has the reduction of the census curve 2.2.2.6.b (s = 4) at 13, where J(𝔽₁₃) ≅ [2,2,6,6]; E₁[3] contains μ₃ and 13 ≡ 1 (mod 3), so a rational point of order 3 on E₁ would give J(𝔽₁₃)[3] rank ≥ 3, and J(ℚ)_tors is exactly [2,2,2,6]; the G2-invariants of the members differ (family_2226_split_certificate.log)", "inherited_from": null } } }, { "key": "2.2.2.8", "group": [ 2, 2, 2, 8 ], "bracket": "[2,2,2,8]", "label": "ℤ/2 ⊕ ℤ/2 ⊕ ℤ/2 ⊕ ℤ/8", "order": 64, "rank": 4, "classes": { "simple": [ { "id": "2.2.2.8.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "463250390625", "463480444900", "230082726", "28452", "1" ], "h": [ "0" ], "equation": { "f": "x^5 + 28452*x^4 + 230082726*x^3 + 463480444900*x^2 + 463250390625*x", "h": "0", "text": "y^2 = x^5 + 28452*x^4 + 230082726*x^3 + 463480444900*x^2 + 463250390625*x" }, "conductor": "71089200", "conductor_sort": 71089200, "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", "discriminant_digits": 75, "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": "" }, "sources": [ "BNSS2026" ], "rm": false, "discovered_by": "Balakrishnan–Najman–Shnidman–Sutherland 2026", "credit_differs": false, "year": null, "historical": false, "first_source": "BNSS2026", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 70 (source: new)", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 41 is strict (χ = T^4 + 4*T^3 + 6*T^2 + 164*T + 1681)", "verified": "2026-09-18 17:33:04 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "2.2.2.8.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "-45", "42", "30", "-23", "-9", "3", "1" ], "h": [ "0", "1", "1" ], "equation": { "f": "x^6 + 3*x^5 - 9*x^4 - 23*x^3 + 30*x^2 + 42*x - 45", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = x^6 + 3*x^5 - 9*x^4 - 23*x^3 + 30*x^2 + 42*x - 45" }, "conductor": "4410", "conductor_sort": 4410, "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)", "discriminant_digits": 15, "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" }, "sources": [ "BookerSutherland" ], "rm": false, "discovered_by": "Booker–Sutherland (extended database)", "credit_differs": false, "year": null, "historical": false, "first_source": "BookerSutherland", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 75 (source: \\cite{BookerSutherland})", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:41:15 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "BNSS2026", "qsplit": "BookerSutherland", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 2, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite", "BNSS2026" ], "note": "the rational curve on the K3 surface M(2,2,2,8) of BNSS2026, Theorem 4.7: (a,b,c,d) = (4t²(t+1)/(t²+t+1)², t/(t+1), −1, −t) on y² = x(x+a²)(x+b²)(x+c²)(x+d²). A rational point P has 4[P − ∞] = [x, 0] ≠ 0 and 8[P − ∞] = 0, and with three classes of order 2 it marks [2,2,2,8] over ℚ(t). Every t ≡ 2 (mod 403) has #J(𝔽₁₃) = 256 and #J(𝔽₃₁) = 1088 (gcd 64), strict at 31 (family_2228_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "bielliptic curves with six rational Weierstrass points and the quotient E = 210e2 (torsion ℤ/2 × ℤ/8) fixed (the Howe–Leprévost–Poonen (2,2)-gluing, HLP 2000, of E and E₂(m), with full gain): C_m: y² = −m(m − 1)(175m − 256)(x² − r₁²)(x² − r₂²)(x² − r₃²), r₁ = m² − 256/175, r₂ = m² − 2m + 256/175, r₃ = −m² + 512m/175 − 256/175 (the lines through (1 : 1 : 1) on the conic r₃² − r₁² = (256/175)(r₂² − r₁²)); (x, y) ↦ (μx² + k, μy/s) maps C_m onto E: Y² = (u + 36)(u − 31/4)(u − 28) and (x, y) ↦ (1/x², y/x³) onto E₂(m), so J is isogenous over ℚ to E × E₂(m). Over ℚ(m), J[2] is rational and the pullback of the point (4, −62) of order 8 has order 8, so [2,2,2,8] is marked. E₂(m) has the same 2-Kummer images as E, so it contains ℤ/2 × ℤ/4 and #J(𝔽_p) is always divisible by 128; the group structure is used instead: every 11- and 13-integral m ≡ 8/5 (mod 143) has the reductions of m = 8/5 (the census curve 2.2.2.8.b, with E₂ = 21a1) at 11 and 13, where J(𝔽₁₁) ≅ [2,2,2,16] (no odd part) and J(𝔽₁₃) ≅ [2,2,8,8] (2-primary exponent 8), so J(ℚ)_tors is exactly [2,2,2,8]; the G2-invariants of the members differ (family_2_2_2_8_split_certificate.log)", "inherited_from": null } } }, { "key": "2.2.2.10", "group": [ 2, 2, 2, 10 ], "bracket": "[2,2,2,10]", "label": "ℤ/2 ⊕ ℤ/2 ⊕ ℤ/2 ⊕ ℤ/10", "order": 80, "rank": 4, "classes": { "simple": [ { "id": "2.2.2.10.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "-2275", "191", "3955", "-767", "-1680", "576" ], "h": [ "0" ], "equation": { "f": "576*x^6 - 1680*x^5 - 767*x^4 + 3955*x^3 + 191*x^2 - 2275*x", "h": "0", "text": "y^2 = 576*x^6 - 1680*x^5 - 767*x^4 + 3955*x^3 + 191*x^2 - 2275*x" }, "conductor": "319360204800", "conductor_sort": 319360204800, "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", "discriminant_digits": 37, "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": "" }, "sources": [ "Elkies2024" ], "rm": false, "discovered_by": "Elkies 2024", "credit_differs": false, "year": null, "historical": false, "first_source": "Elkies2024", "first_source_year": 2024, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 71 (source: \\cite{Elkies2024})", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 11 is strict (χ = T^4 - 4*T^3 + 6*T^2 - 44*T + 121)", "verified": "2026-09-18 17:33:05 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "Elkies2024", "qsplit": null, "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": false, "gsplit": false, "undecided": false }, "exclusions": { "qsplit": { "reason": "A Jacobian split over ℚ contains an elliptic curve over ℚ with an elliptic quotient over ℚ. By Mazur's theorem, a rational point of order 7 then forces rational 2-rank at most 2, and rational 2-rank 4 excludes rational points of orders 5 and 7.", "sources": [ "QSplitTwoRankObstruction", "Mazur1977" ], "proof_url": "proofs/qsplit-two-rank-obstruction.html" } }, "n_curves": 1, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite", "Elkies2024" ], "note": "Elkies' Clebsch–Klein family y² = x∏(x − rᵢ²), Σrᵢ = Σrᵢ³ = 0, on the tangent section of the Clebsch surface at P₀ = (1, −8, −7, 5, 9): r(u) = (Σvᵢ³)P₀ − 3(ΣP₀ᵢvᵢ²)v with v = (7, 56u, 0, −10 − 17u, 3 − 39u). [∞₊ − ∞₋] has order 5 and J[2] is rational, so [2,2,2,10] is marked over ℚ(u). Every 11-integral u ≡ 7 (mod 11) has #J(𝔽₁₁) = 80, strict (family_22210_certificate.log)", "inherited_from": null }, "split": { "grade": "open", "sources": [], "note": "", "inherited_from": null } } }, { "key": "2.2.2.12", "group": [ 2, 2, 2, 12 ], "bracket": "[2,2,2,12]", "label": "ℤ/2 ⊕ ℤ/2 ⊕ ℤ/2 ⊕ ℤ/12", "order": 96, "rank": 4, "classes": { "simple": [ { "id": "2.2.2.12.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "-57510204598294806758400", "9819234969339333136", "-603825803380840", "16672898913", "-211210", "1" ], "h": [ "0" ], "equation": { "f": "x^6 - 211210*x^5 + 16672898913*x^4 - 603825803380840*x^3 + 9819234969339333136*x^2 - 57510204598294806758400*x", "h": "0", "text": "y^2 = x^6 - 211210*x^5 + 16672898913*x^4 - 603825803380840*x^3 + 9819234969339333136*x^2 - 57510204598294806758400*x" }, "conductor": "4778692469215148637760180473900", "conductor_sort": 4778692469215148637760180473900, "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", "discriminant_digits": 140, "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": "" }, "sources": [ "BNSS2026" ], "rm": false, "discovered_by": "Balakrishnan–Najman–Shnidman–Sutherland 2026", "credit_differs": false, "year": null, "historical": false, "first_source": "BNSS2026", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 72 (source: new)", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 71 is strict (χ = T^4 - 8*T^3 + 46*T^2 - 568*T + 5041)", "verified": "2026-09-18 17:33:12 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "BNSS2026", "qsplit": null, "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": false, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 1, "infinite": { "simple": { "grade": "exact", "sources": [ "ThisSite", "Audit2026", "BNSS2026", "Choudhry" ], "note": "BNSS2026, Theorems 3.1 and 3.3 give the family C_P, P = ψ(nQ) on the surface a²+b²+c² = u²+v², a⁴+b⁴+c⁴ = u⁴+v⁴, with E: y² = x³ − 21x − 20 of rank 1 and ψ(2Q) = P₁ = [120:143:266:218:241]. For n ≡ 2 (mod 560) the curves C_{ψ(nQ)} have the same reduction as C_{P₁} at 29, 37 and 71, so their torsion divides gcd(#J(𝔽₂₉), #J(𝔽₃₇)) = 96 = #[2,2,2,12] and their Frobenius at 71 is strict; the G2-invariants at ψ(2Q), ψ(4Q) differ (family_22212_certificate.log; argument from the audit of 2026-09-18). Since 2026-10-18 the certificate also checks the inputs from BNSS2026 over the function field of E: ψ(E) lies on the surface, and explicit classes generate [2,2,2,12] in J(ℚ(E))", "inherited_from": null }, "split": { "grade": "open", "sources": [], "note": "", "inherited_from": null } } }, { "key": "2.2.2.16", "group": [ 2, 2, 2, 16 ], "bracket": "[2,2,2,16]", "label": "ℤ/2 ⊕ ℤ/2 ⊕ ℤ/2 ⊕ ℤ/16", "order": 128, "rank": 4, "classes": { "simple": [], "qsplit": [ { "id": "2.2.2.16.a", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "230262357769227286812882682545067236", "-80303936093332009982025661643142", "-406358837350345916288092715922", "249323508190639120726354536", "9716619250580389984680", "-42337414886067900" ], "h": [ "0", "1", "1" ], "equation": { "f": "-42337414886067900*x^5 + 9716619250580389984680*x^4 + 249323508190639120726354536*x^3 - 406358837350345916288092715922*x^2 - 80303936093332009982025661643142*x + 230262357769227286812882682545067236", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = -42337414886067900*x^5 + 9716619250580389984680*x^4 + 249323508190639120726354536*x^3 - 406358837350345916288092715922*x^2 - 80303936093332009982025661643142*x + 230262357769227286812882682545067236" }, "conductor": "1328257075878182563154683193645100", "conductor_sort": 1328257075878182563154683193645100, "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)", "discriminant_digits": 226, "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": "" }, "sources": [], "rm": false, "discovered_by": "F. Najman", "credit_differs": false, "year": 2026, "historical": false, "first_source": null, "first_source_year": null, "torsion_method": "generators", "reference": "new", "new_group": true, "new_for_class": true, "certificate": "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", "verified": "2026-10-16 07:12:43 UTC", "submitted": "2026-10-16", "source_kind": "issue" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": 0, "gsplit": null, "undecided": null }, "first_source": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "known": { "simple": false, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4) of E₁ = E_{2,4}(t) with E₂ = E_{2,8}(−20/31) along the identity on the labelled 2-torsion, on the fibre t = Aa², t + 1/4 = Bc², t − 1/4 = Ce², (A, B, C) = (−1788961, 11195, −138818), of the locus where the Kummer images of the rational torsion of E₁ and E₂ agree completely; this fibre is a 2-covering of E_d: y² = x³ − d²x, d = 554577910 (rank 1). C_t: y² = −A(t)(k₁t − x²)(k₂(t − 1/4)² − x²)(x² + k₃(t + 1/4)²) maps onto E₂ by (x, y) ↦ (x², y) and onto E₁(t) by (x, y) ↦ (1/x², y/x³), so J is isogenous over ℚ to E₁ × E₂. [2,2,2,16] is marked over the function field of the fibre: the class of order 16 is the sum of the pullbacks of halves of the points of order 4 and 8, computed over ℚ(√22390, √69409)(n) and descending to ℚ(n). For R = P + (d, 0) + 12kP on E_d (P of order 6 mod 11 and 4 mod 13) the curve C_{x(R)/4d} has the reductions of the census curve 2.2.2.16.a (k = 0) at 11 and 13, where J(𝔽₁₁) ≅ J(𝔽₁₃) ≅ [2,2,2,16], so J(ℚ)_tors is exactly [2,2,2,16]; the G2-invariants at k = 0 and at 3P + (d, 0) differ (family_22216_split_certificate.log)", "inherited_from": null } } }, { "key": "2.2.4.4", "group": [ 2, 2, 4, 4 ], "bracket": "[2,2,4,4]", "label": "ℤ/2 ⊕ ℤ/2 ⊕ ℤ/4 ⊕ ℤ/4", "order": 64, "rank": 4, "classes": { "simple": [ { "id": "2.2.4.4.a", "class": "simple", "class_certified": true, "status": "certified", "class_text": "geometrically simple", "f": [ "0", "300512487407616", "414985109760", "173387808", "26065", "1" ], "h": [ "0" ], "equation": { "f": "x^5 + 26065*x^4 + 173387808*x^3 + 414985109760*x^2 + 300512487407616*x", "h": "0", "text": "y^2 = x^5 + 26065*x^4 + 173387808*x^3 + 414985109760*x^2 + 300512487407616*x" }, "conductor": "53430369300", "conductor_sort": 53430369300, "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", "discriminant_digits": 79, "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": "" }, "sources": [ "BNSS2026" ], "rm": false, "discovered_by": "Balakrishnan–Najman–Shnidman–Sutherland 2026", "credit_differs": false, "year": null, "historical": false, "first_source": "BNSS2026", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 1, row 73 (source: new)", "new_group": true, "new_for_class": true, "certificate": "geometrically simple: Frobenius at p = 37 is strict (χ = T^4 - 4*T^3 - 2*T^2 - 148*T + 1369)", "verified": "2026-09-18 17:33:12 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "qsplit": [ { "id": "2.2.4.4.b", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "4913", "867", "-5657", "-671", "1000", "60" ], "h": [ "0", "1", "1" ], "equation": { "f": "60*x^5 + 1000*x^4 - 671*x^3 - 5657*x^2 + 867*x + 4913", "h": "x^2 + x", "text": "y^2 + (x^2 + x)*y = 60*x^5 + 1000*x^4 - 671*x^3 - 5657*x^2 + 867*x + 4913" }, "conductor": "485100", "conductor_sort": 485100, "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)", "discriminant_digits": 33, "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": "" }, "sources": [ "BNSS2026" ], "rm": false, "discovered_by": "Balakrishnan–Najman–Shnidman–Sutherland 2026", "credit_differs": false, "year": null, "historical": false, "first_source": "BNSS2026", "first_source_year": 2026, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 76 (source: new)", "new_group": false, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:41:17 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": "BNSS2026", "qsplit": "BNSS2026", "gsplit": null, "undecided": null }, "known": { "simple": true, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 2, "infinite": { "simple": { "grade": "exact", "sources": [ "Najman2026", "ThisSite", "BNSS2026" ], "note": "the rational curve in A(2,2,4,4) of BNSS2026, Proposition 4.4, on y² = x(x+a²)(x+b²)(x+c²)(x+d²): a half D₀ of [x, 0], a half D₁ of [(x+a²)(x+b²), 0], [x+a², 0] and [x+c², 0] mark [2,2,4,4] over ℚ(s). Every s ≡ 3 (mod 1591) has #J(𝔽₃₇) = 1216 and #J(𝔽₄₃) = 1856 (gcd 64), strict at 37 (family_2244_certificate.log)", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4) of E₁ = E_{2,4}(t) with E₂ = E_{2,4}(u) along the identity on the labelled roots of 2-torsion (special root first), on the locus where the Kummer images of the rational torsion agree completely: tu, (t + 1/4)(u + 1/4) and (t − 1/4)(u − 1/4) are squares. Over the t-line its fibre is the genus-1 curve u = ta², u + 1/4 = (t + 1/4)b², u − 1/4 = (t − 1/4)c²; with origin (1, 1, 1) the point R = (−1, 1, 1) has infinite order, and 2R gives u = (t⁹/9 + t⁷/12 + 5t⁵/384 − t³/1024 + t/65536)/(t⁸ − t⁶/4 + 5t⁴/384 + t²/3072 + 1/589824). Up to a square C_t: y² = F(x²), with (x, y) ↦ (x², y) onto E₂ and (x, y) ↦ (1/x², y/x³) onto E₁, so J is isogenous over ℚ to E₁ × E₂. Over ℚ(t) J[2] is rational, the pullback D₄ of the point of order 4 has order 4, and the gain G₄ = pullback of a half of P₁ plus pullback of a half of P₂, computed over ℚ(t)(√(2(4t + 1)), √(2(1 − 4t))), has coefficients in ℚ(t), order 4 and 2G₄ = D₄ + D₄′; together they generate [2,2,4,4] (64 elements at t = 6). Every 13- and 19-integral t ≡ 6 (mod 247) has the reductions of t = 6 at 13 and 19, where J(𝔽₁₃) ≅ [2,2,8,8] and J(𝔽₁₉) ≅ [2,2,12,12], so J(ℚ)_tors is exactly [2,2,4,4]. The census curve 2.2.4.4.b is the gluing at the locus point (t, u) = (15/4, 135/484) (not on this section); the G2-invariants of the members differ (family_2_2_4_4_split_certificate.log)", "inherited_from": null } } }, { "key": "2.2.4.8", "group": [ 2, 2, 4, 8 ], "bracket": "[2,2,4,8]", "label": "ℤ/2 ⊕ ℤ/2 ⊕ ℤ/4 ⊕ ℤ/8", "order": 128, "rank": 4, "classes": { "simple": [], "qsplit": [ { "id": "2.2.4.8.a", "class": "qsplit", "class_certified": true, "status": "certified", "class_text": "split over ℚ", "f": [ "0", "42938243694863192676456480983270400000000000000", "558098950720994323170545736900000000", "1821369625722709924500000", "2288161182841", "1" ], "h": [ "0" ], "equation": { "f": "x^5 + 2288161182841*x^4 + 1821369625722709924500000*x^3 + 558098950720994323170545736900000000*x^2 + 42938243694863192676456480983270400000000000000*x", "h": "0", "text": "y^2 = x^5 + 2288161182841*x^4 + 1821369625722709924500000*x^3 + 558098950720994323170545736900000000*x^2 + 42938243694863192676456480983270400000000000000*x" }, "conductor": "1584038398842900", "conductor_sort": 1584038398842900, "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)", "discriminant_digits": 233, "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": "" }, "sources": [ "HLP2000" ], "rm": false, "discovered_by": "Balakrishnan–Najman–Shnidman–Sutherland 2026", "credit_differs": true, "year": null, "historical": false, "first_source": "HLP2000", "first_source_year": 2000, "torsion_method": "TorsionSubgroup", "reference": "Balakrishnan–Najman–Shnidman–Sutherland 2026, Table 2, row 77 (source: \\cite{HLP2000})", "new_group": true, "new_for_class": true, "certificate": "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", "verified": "2026-09-18 17:41:19 UTC", "submitted": "2026-08-30", "source_kind": "import" } ], "gsplit": [], "undecided": [] }, "first_dated": { "simple": null, "qsplit": null, "gsplit": null, "undecided": null }, "first_source": { "simple": null, "qsplit": "HLP2000", "gsplit": null, "undecided": null }, "known": { "simple": false, "qsplit": true, "gsplit": false, "undecided": false }, "exclusions": {}, "n_curves": 1, "infinite": { "simple": { "grade": "open", "sources": [], "note": "", "inherited_from": null }, "split": { "grade": "exact", "sources": [ "ThisSite", "HLP2000" ], "note": "the Howe–Leprévost–Poonen (2,2)-gluing (HLP 2000, Prop. 4 and §3.7) of E₁ = E_{2,8}(t) with E₂ = E_{2,8}(u) along the identity on the 2-torsion points labelled as in HLP Table 5, where (t, u) is the section 2σ of HLP's elliptic surface (5), X(t)y² = X(u), X = (8u² − 1)(8u² + 8u + 1), σ = (t, −1), zero section (t, 1) (u of degree 9 in t), so a half of (T₂, T₂′) gives a new rational point of order 2 (HLP Prop. 12). Up to a square C_t: y² = F(x²), with (x, y) ↦ (x², y) onto E₂ and (x, y) ↦ (1/x², y/x³) onto E₁, so J is isogenous over ℚ to E₁ × E₂. Over ℚ(t) the sextic splits into linear factors (J[2] rational), and with the pullbacks of the points of order 8 it generates [2,2,4,8] (128 elements at t = 4). Here #J(𝔽_p) is always divisible by 256, so the group structure is used: every 13- and 41-integral t ≡ 4 (mod 533) has the reductions of t = 4 at 13 and 41, where J(𝔽₁₃) ≅ [2,2,8,8] and J(𝔽₄₁) ≅ [2,2,4,96], so J(ℚ)_tors is exactly [2,2,4,8]; the G2-invariants of the members differ (family_2_2_4_8_split_certificate.log)", "inherited_from": null } } } ] }