About — Torsion of genus 2 Jacobians over ℚ

About the census

This site keeps a verified list of the finite abelian groups known to occur as the rational torsion subgroup J(ℚ)_tors of the Jacobian J of a genus 2 curve over ℚ, with a certified example for each group in each of three classes of Jacobians. It is maintained by Filip Najman (University of Zagreb); the data, the verification code and every certificate live in the GitHub organisation Genus-2-torsion. It is seeded with Tables 1 and 2 of Balakrishnan–Najman–Shnidman–Sutherland, Rational torsion on simple genus two Jacobians (2026), taken from the machine-readable tables of its certification repository. Contributions of new curves are welcome — see submit a curve.

Conventions

the curve: A smooth projective curve X of genus 2 over ℚ, given by an affine model y² + h(x)y = f(x) with deg f ≤ 6, deg h ≤ 3. Curves are recorded up to isomorphism over ℚ; the verifier works on the even model y² = 4f + h², made integral, and records the reduced minimal Weierstrass model and the conductor. For a Jacobian isogenous over ℚ to a product E₁ × E₂ found by the Richelot search, the conductor is N(E₁)N(E₂) (rigorous: an isogeny invariant, with the elliptic conductors from Tate's algorithm). Otherwise the conductor exponent at 2 is computed by Magma with Ogg's formula when v₂(Δ) ≥ 12, without a correctness guarantee — an audit found it too large by 2¹⁰ or 2²⁰ for nine split curves — and this is flagged on the curve's page. Odd exponents are rigorous. When Magma's genus-2 conductor fails or does not finish, or when its exponent at 2 disagrees with the LMFDB label while the odd parts agree, the conductor is read off the LMFDB label. The one curve with neither, 11.b, has no conductor yet.
the group: J(ℚ)_tors, written as a list of invariant factors [n₁, …, n_r] = ℤ/n₁ ⊕ … ⊕ ℤ/n_r with n₁ | … | n_r, [ ] for the trivial group, as in the paper. The groups are listed in the paper's order: by number of invariant factors, then lexicographically.
the three classes: Every curve is placed in exactly one of: geometrically simple (J is isogenous to a product of elliptic curves over no extension of ℚ); split over ℚ (J is isogenous over ℚ to a product of two elliptic curves over ℚ); simple over ℚ but split over ℚ̄ (for instance the Weil restriction of an elliptic curve over a quadratic field, or J₁(13)). The paper's Table 1 is the first class, its Table 2 the union of the other two. A curve enters a class only with a certificate; otherwise it is listed as "class not certified", with the evidence.
which curves are recorded: For each group and class the census keeps the known curve of smallest conductor: a submitted curve is accepted when its (group, class) is new, or when its conductor is strictly smaller than that of every census curve with the same group and class (a curve whose conductor cannot be computed is then rejected, as it cannot be compared). Earlier curves stay in the census, so the group page shows the history of the record. In addition the curators may enter the earliest known curve of a group with its discovery year ("historical example"), whatever its conductor. A curve isomorphic over ℚ to a census curve is a duplicate and rejected; geometrically isomorphic curves (twists) are accepted and cross-referenced.
smallest conductor / first example: The home page shows, for each group and class, either the curve of smallest conductor or — with the toggle — the earliest curve with a recorded discovery year. The seed curves have no year of their own: the paper's tables give the smallest-conductor example and credit the first source of a realisation of the group, which need not be that curve (for [5] it credits Ogg 1973, whose curve is a modular curve of larger conductor). In the "first example" view such cells show the smallest-conductor curve dimmed, with "first known: source (year)"; submitting the original curves is welcome. Each seed curve's own credit was compared with the curves printed in the cited sources and with the LMFDB on 14 and 20 October 2026; the corrections, and the credits that could not be checked, are listed in data/knowledge/sources/PROVENANCE.md.

How a curve is certified

Every submission is verified from scratch by pipeline/magma/verify_lib.m on Mordell, with Magma V2.29-4 (the 160 curves verified up to 16 October 2026) or V2.30-1 (all later verifications); each curve's page records the version used.

Torsion. First the gcd of #J(𝔽_p) over 25 good primes p ≥ 3 is computed (reduction is injective on torsion, so #J(ℚ)_tors divides it). If the submitter gave generators — points of the Jacobian of the even model y² = 4f + h² in Mumford representation — Magma checks that they are torsion points and independent (all sums of multiples are distinct); if the order of the group they generate equals the gcd, that group is J(ℚ)_tors and no search is needed (method "generators"). Otherwise J(ℚ)_tors is computed exactly with Magma's TorsionSubgroup (Stoll's algorithm with the Müller–Stoll height bounds), and its generators are recorded. The Frobenius polynomials behind the gcd and the certificates below are computed by brute-force point counting (Al := "Naive"): Magma V2.29-4's default algorithm returns a wrong polynomial for a small proportion of genus 2 curves over 𝔽₃, 𝔽₅, 𝔽₇; every polynomial is cross-checked against Magma's point count, the functional equation and the Weil bounds.
Geometric simplicity. A good prime p is strict if the characteristic polynomial χ_p of Frobenius on J mod p is irreducible over ℚ and, for a root π, no power πⁿ with n ≤ 12 generates a proper subfield of ℚ(π). One strict prime proves that J is geometrically simple: if J became isogenous to a product over an extension of ℚ, so would its reduction over an extension 𝔽_pⁿ, which forces a ratio of two Frobenius eigenvalues to be a root of unity of order dividing n; such a root of unity lies in the Galois closure of the quartic CM field ℚ(π), a field of degree 4 or 8 whose Galois group is C₄, V₄ or D₄, and the roots of unity in such a field have order at most 12 (orders 5, 8 and 10 occur, and are covered because every n ≤ 12 is tested). This is the criterion of verify_simple_certificates.m in the paper's repository (Leprévost's root-power criterion). Primes up to 500 are tried.
Simplicity over ℚ. If J ~ E × E′ over ℚ then χ_p = χ_E,p·χ_E′,p is reducible at every good prime, so one irreducible χ_p proves that J is simple over ℚ.
Splitness. A certificate is an explicit geometric object, checked exactly: an involution of the curve over ℚ other than the hyperelliptic one (its quotient has genus 1, so J is split over ℚ); a chain of at most two Richelot (2,2)-isogenies over ℚ (Magma's RichelotIsogenousSurfaces) reaching a product of two elliptic curves over ℚ (split over ℚ) or the Weil restriction of an elliptic curve over a quadratic field (geometrically split); more than one involution in the geometric automorphism group (bielliptic over ℚ̄); such chains over quadratic fields ℚ(√d); or a map (x, y) ↦ (p/q, y·hh/q²) to a genus 1 curve supplied with the submission (the 18 covers of the paper's Table 2 that are not reached by Richelot isogenies were transcribed from verify_split_certificates.m), verified by the polynomial identity (e₃p³ + e₂p²q + e₁pq² + e₀q³)·q = g·hh² with a nonzero Wronskian. A geometrically split Jacobian is placed in the class "split over ℚ" when a certificate over ℚ exists, and in "simple over ℚ, split over ℚ̄" when it is simple over ℚ by 3.
Evidence, not proof. When no certificate is found, the curve's page records the split signature: whether every good prime below 200 fails the strictness test. A geometrically split Jacobian always fails it, so this is strong evidence of splitness, but finitely many Frobenius polynomials never prove that an endomorphism exists.

Consistency checks abort the verification if both a simplicity and a splitness certificate are found, or if a certificate over ℚ coexists with an irreducible χ_p. The torsion subgroups, geometric simplicity and geometric splitness of all 150 seed curves were re-derived by this pipeline and agree with the paper's tables; the class over ℚ of the split curves is new information computed here.

Impossible in a class

A dash in an example column means that no example is recorded. The label impossible means that a proof excludes that torsion group in the indicated class, and links to the proof. For Jacobians split over ℚ, Mazur's theorem applied to an elliptic subvariety and its elliptic quotient excludes rational torsion of every prime order greater than 7. This marks 14 census groups as impossible in that column. The result concerns splitting over ℚ; the census has geometrically split examples with torsion [11] and [19] that are simple over ℚ. The same argument shows that a rational point of order 7 forces rational 2-rank at most 2, and that rational 2-rank 4 excludes points of order 5 and 7. This excludes [2,2,14] and [2,2,2,10] in that column.

A geometrically split abelian surface over ℚ has no rational point of order 17, 22, 23, 26, 29, 31, 33, 34, or 39. Complete proofs are available for [17], [22] and [2,22], [23] and [29], [26], [2,26] and [39], [31], and [33] and [34], each with downloadable LaTeX. The proofs combine elliptic torsion bounds with the Galois action on the geometric endomorphism algebra and apply to every geometric splitting degree. Both split example columns are therefore labelled impossible for these eleven groups, and their combined geometrically split ∞? cells show ∅, linked to the corresponding proof.

A geometrically split genus-2 Jacobian with a rational point of order 13 is not simple over ℚ (proof). Together with Mazur's theorem in the split-over-ℚ column, this makes [13] impossible on the whole geometrically split side, and its ∞? cell shows ∅.

On a geometrically split abelian surface that is simple over ℚ, a rational point of order 7 forces rational 2-torsion of rank at most 2 (proof). Together with the 2-rank obstruction in the split-over-ℚ column, this makes [2,2,14] impossible on the whole geometrically split side, and its ∞? cell shows ∅.

The following groups are impossible only in the “geometrically split, simple over ℚ” column: [30], [60] and [2,30] (no point of order 30), [35] and [70] (no point of order 35), [63] (no point of order 63), and [5,5], [5,10] and [7,7] (rational p-torsion of rank at most 1 for p ≥ 5). Simplicity over ℚ is essential: each of these groups has examples in the split-over-ℚ class. Thus these exclusions do not imply ∅ on the combined geometrically split side.

Exclusions beyond the census records six further exact groups, [4,64], [8,32], [16,16], [2,2,64], [2,4,32], and [2,8,16], excluded for Jacobians split over ℚ. A theorem of 11 November 2026 shows that rational torsion of order 256 forces full rational 2-torsion in this class, at every splitting degree. These six groups have no census rows; the result makes no claim about the other two classes.

Infinitely many?

For each group and each side of the census (geometrically simple; geometrically split) the census answers "are infinitely many such Jacobians known?" with one of the following four labels. The label ∅ excludes the group on the indicated side of the census.

∞: Certified, exact. Infinitely many pairwise non-isomorphic curves in the class whose Jacobian has torsion subgroup exactly the group. On the simple side this follows the three-part standard of the record Najman 2026 (unpublished): (L1) a family over a rational base with a marked subgroup isomorphic to G over the function field ℚ(t), and one fiber t₀ with torsion exactly G; (L2) a strict prime p for that fiber; (L3) fibers with pairwise distinct G2-invariants, so the moduli map is not constant. These give infinitely many fibers with torsion exactly G and geometrically simple Jacobian by the following argument (the record's phrase "excess torsion is a closed condition" is not the right justification — rational torsion can jump on a Zariski-dense thin set of parameters — as an audit of 18 September 2026 pointed out): the torsion of the generic fiber injects into J_t₀(ℚ)_tors = G and contains the marked G, so it is exactly G; for a second good prime q of the fiber t₀ and every integer t ≡ t₀ (mod pq), J_t has the same reductions as J_t₀ at p and q, so #J_t(ℚ)_tors divides B = gcd(#J_t₀(𝔽_p), #J_t₀(𝔽_q)) and J_t is geometrically simple (same strict Frobenius polynomial at p); the t with a rational torsion point of some order m | B outside G lie in finitely many thin sets, which by Hilbert irreducibility miss infinitely many integers of the residue class; and the moduli map has finite fibers. The families and their fiber certificates are listed in the record (copied to data/knowledge/sources/census_infinity_record.md, with the Magma scripts in the private repository it comes from). Since 18 October 2026 every exact grade on the simple side also has an explicit residue-class certificate computed for this site (family_<group>_certificate.log in data/knowledge/sources/, shared argument in pipeline/magma/family_cert_lib.m): the marked group is checked over the function field, and on an explicit residue class the gcd of #J(𝔽_p) over one or two primes equals #G (for [4,4] and [2,8], whose orders are always divisible by 64, the group structures of J(𝔽_p) at three primes leave only G), with a strict prime, so every member of the class has torsion exactly G and geometrically simple Jacobian, without Hilbert irreducibility. For the 36 groups whose grade rested on the record alone this was done by an audit of the record (18 October 2026), on the record's own families (one-parameter slices where the record's chart has more parameters), except that the five-torsion chart the record refers to for [5] has torsion containing [10], so [5] is certified with y² = (1+x+x²)² − tx⁵; the record's plain M(12) chart for [6] has torsion exactly [6] generically but no residue class caps its 2-part, so [6] is certified with a contact-6 family; and the record's "Richelot partner #4" for [2,8] denotes two families, of which the one through t = 2 is certified (see the errata in PROVENANCE.md). The certificates for [ ], [2], [13], [15], [16], [17], [19], [24], [25], [26], [2,14], [2,16], [2,2,8], [2,2,14], [2,4,4], [2,2,2,12] and [3,12] were computed earlier for this site (the record's family for [2] has no rational 2-torsion generically; these 17 certificates were audited on 18 October 2026, which moved [2] to a residue class with gcd 2 and added checks of the inputs from BNSS to [2,2,2,12]. For [13], [15], [16], [17] and [19] a single prime suffices, since B = #J(𝔽₃) = #G on a residue class; the families for these and for [24], [25], [26], [2,16] come from relations between two rational points, as for [19]; the family for [2,4,4] is Nicholls' family Λ₃₃₄ on its rational locus where a quadratic factor splits. For [2,2,14], [2,2,2,12] and [3,12] the base is an elliptic curve of rank 1, where the same congruence argument applies directly). The record also lists [4,8] as certified, but no fiber certificate exists for its family; it was graded open here from 13 to 19 October 2026, when a family on a rational curve of the same tangent cover was certified (family_4_8_certificate.log); over a rank-1 elliptic curve on that rational curve a quadratic factor splits, which gives [2,4,8] (family_2_4_8_certificate.log). The family for [28] is the one of Platonov–Fedorov (2018), with a rational point of order 28 over ℚ(t); it was transcribed and certified here on 19 October 2026 (family_28_certificate.log). The literature source of a family is cited next to the record. The three groups that had only inherited a family from a larger group, [3,6], [2,2,6] and [2,2,12], got their own families on 18 October 2026 (family_3_6_certificate.log, family_2_2_6_certificate.log, family_2_2_12_certificate.log): Bruin–Flynn–Testa's (ℤ/3)² family on a rational curve along which the sextic acquires a rational root; y² = F₁F₂ with F₁ − F₂ constant (so ∞₊ − ∞₋ has order 3) and four rational roots; and the same mechanism on the curves of BNSS with one square condition relaxed, where the irrational pair of roots still has a rational half. A second [2,2,12] family over a rank-1 elliptic curve passes through the census curve 2.2.12.a (family_2_2_12_control_certificate.log). The group [3,9], graded open until 19 October 2026 (only isolated examples were known), has a family on a line of the [9]-chart: a pencil with a class of order 9 over the function field, on which the base change t = (s² − 1)/4 makes an independent class of order 3 rational (family_3_9_certificate.log). On the split side 70 groups are certified exact. For [8,8] (family_88_split_certificate.log) the Jacobians are split by explicit maps onto an elliptic curve, the group is marked over the function field, and since #J(𝔽_p) is always divisible by 256 in that family, exactness on a residue class comes from the group structures of J(𝔽_p) at two primes rather than from their orders. For [2,2,2,16] (family_22216_split_certificate.log) the family is a (2,2)-gluing of an elliptic curve with torsion [2,4] and a fixed one with torsion [2,8], parametrised by a genus-1 curve whose Jacobian has rank 1; the group is marked over its function field, and the residue-class argument at 11 and 13 is the one used for the simple groups. For [14], [2,14] and [2,2,10] (family_14_split_certificate.log, family_214_split_certificate.log, family_2210_split_certificate.log) the families are (3,3)-gluings of a fixed elliptic curve with torsion [7] or [10] with partners carrying the required 2-torsion, split by explicit degree-3 maps and parametrised by a rational curve resp. a genus-1 curve of rank 1. For [2,10], [2,2,2,6] and [4,12] (family_210_split_certificate.log, family_2226_split_certificate.log, family_412_split_certificate.log) they are (2,2)-gluings of a Kubert curve with an isogenous curve, rational in one parameter (for [4,12] the class of order 4 added by the gluing is computed over a quartic extension of the function field and descends; the census curve 4.12.a is a member); there #J(𝔽_p) is a square, so the odd part is pinned down by a Galois argument (a factor contains μ₅ resp. μ₃) together with the group structure of J(𝔽_p) at a prime p ≡ 1 (mod 5) resp. (mod 3). For [18] and [2,18] (family_18_split_certificate.log, family_218_split_certificate.log) the families are (3,3)-gluings of a fixed elliptic curve with torsion [9] (54b3 resp. 714i1) with partners carrying a rational 2-torsion point resp. full 2-torsion, parametrised by a rational curve resp. a genus-1 curve of rank 1; the partners inherit a rational 3-torsion point, but no extra 3-torsion arises (it would need ℤ/18 on an elliptic curve over ℚ), and since #J(𝔽_p) is always divisible by 54 the group structures of J(𝔽_p) at two resp. three primes cap the torsion. Since 19 October 2026 every other split group recorded as "family" has its own exact family, with one certificate per group (family_<group>_split_certificate.log, e.g. family_2_2_4_split_certificate.log for [2,2,4]). The groups [1]–[10], [12], [2,2], [2,4], [2,6], [2,8], [2,12], [2,2,2], [2,2,4], [2,2,6], [2,2,8] and [2,2,12] come from (3,3)-gluings of a fixed elliptic curve whose torsion is the elliptic part of the group, with partners carrying no level structure, a rational 2-torsion point, or full 2-torsion. [2,2,2,2], [2,2,2,4] and [2,2,2,8] come from bielliptic curves with six rational Weierstrass points and one quotient fixed. [3,3], [5,5], [6,6], [3,9] and [4,4] come from (2,2)-gluings of a curve with an isogenous curve. For [3,6] the same gluing is twisted by the automorphism of the 2-torsion that removes the extra 2-torsion, and the 2-rank is fixed by Frobenius elements acting on the Weierstrass points. [16], [2,16], [2,2,16], [4,8], [2,4,4] and [2,2,4,4] come from (2,2)-gluings with an added class of order 4 or 16, computed over an extension of the function field and descended. The other groups of Howe–Leprévost–Poonen's Table 1 ([20], [21], [30], [35], [40], [45], [60], [3,12], [2,24], [5,10], [6,12], [2,4,8], [2,6,6], [2,2,24], [2,2,4,8]), with [15] and [24], come from explicit versions of their families, restricted to a rational curve or to an elliptic curve of positive rank. Three groups that were graded open, [3,24], [2,30] and [2,48], come from the loci of those families on which HLP's Proposition 12 adds torsion; these loci are genus-1 curves of positive rank. Two more open groups, [27] and [36], come from the (3,3)-families of [9] and [12]: on a genus-1 locus a class of order 27 appears (a gain of the gluing, not a pullback), resp. the partner acquires a rational point of order 9. These loci are étale triple covers of genus-1 curves of rank 1, so the members carrying the extra torsion are the points of a coset of a subgroup of infinite order, and the certificates check the class at several members. For [48] (family_48_split_certificate.log) the family is Howe's (2015, Theorems 3.1 and 3.2): the (2,2)-gluing of the curve 42a1 (torsion ℤ/8) with curves with a point of order 6, over an elliptic curve of rank 2 on which HLP's Proposition 12 adds a point of order 16; the torsion is exactly ℤ/48 because #J(𝔽₅) = 48 is forced on a residue class. In each case the group is marked over the function field (or at every member, by injectivity of pullbacks), and the orders or group structures of J(𝔽_p) at one to three primes cap the torsion on a residue class. The exception is [4,16] (family_416_split_certificate.log, with a complete proof): an HLP (2,2)-gluing of curves with torsion [2,4] and a fixed curve with torsion [2,8], over a rational curve, whose torsion is exactly [4,16] at every member, from the intersection of the torsion Kummer images of the two elliptic factors; the census curve 4.16.a is a member.
∞ ⊇: Family with containment. A positive-dimensional family (infinitely many members in moduli) of curves in the class whose Jacobians have torsion containing the group is proven in the cited source: Theorem 1 / Table 1 of Howe–Leprévost–Poonen (2000) and Howe's order-48 family (2015) on the split side (their Jacobians are isogenous to products of elliptic curves by construction). A group also inherits this grade from any recorded group containing it as a subgroup (a lighter badge). On the simple side no group has this grade since 18 October 2026: every family recorded there has been certified exact, and the groups that only inherited the grade have their own exact families. On the split side no group has it since 19 October 2026 either: the last one, [48], is certified exact on Howe's family. Whether infinitely many members have torsion exactly the group (and, on the simple side, are geometrically simple) is open.
?: Open. No positive-dimensional construction is recorded — this is not a claim that none exists; the group page gives the reason when the record has one (e.g. only isolated examples, or every known example has real multiplication).
∅: Impossible. A proof excludes every Jacobian on this side with the given torsion group. The badge links to that proof. This currently applies to [13], [22], [23], [26], [29], [31], [33], [34], [39], [2,22], and [2,26] on the geometrically split side.

LMFDB links

A curve in the production LMFDB (conductor ≤ 10⁶, discriminant ≤ 10⁶) is linked by its label, a permalink. A curve of the extended Booker–Sutherland database, available in the alpha LMFDB, is linked by its equation (the LMFDB looks it up, matching any isomorphic model), because labels there are not yet permanent; the snapshot label of the paper (2026-08) is kept as provenance. Other curves get a link that searches the LMFDB by equation.

Sources

Data last built: ….

Workflow and reproducibility

Submissions arrive as GitHub issues (or pull requests). A script on Mordell pulls them, runs the Magma verifier under a time and memory limit (one job at a time), writes one JSON certificate per accepted curve to data/curves/ together with the Magma logs in data/logs/, rebuilds data/groups.json and data/curves.json, and pushes to GitHub, where this page is served. Each certificate records the Magma version, the CPU time, the SHA-256 of the verification code, the strict prime and Frobenius polynomial or the splitness certificate, the generators of the torsion subgroup, and a Magma snippet that reproduces the torsion computation. The whole repository, including rejected submissions with the reason, is public.

How to cite

Please cite the original reference of a curve (given on its page) and, for the census itself, the repository Genus-2-torsion/Genus-2-torsion.github.io together with Balakrishnan–Najman–Shnidman–Sutherland, Rational torsion on simple genus two Jacobians, arXiv:2608.28543. The design of this site follows the census of sporadic points on X₁(m,n) and the Elliptic Curve Rank Leaderboard.