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 when Magma can factor the discriminant quickly. 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.
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.

How a curve is certified

Every submission is verified from scratch by pipeline/magma/verify_lib.m (Magma V2.29) on Mordell.

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.

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 three grades. There is no "finitely many" grade: no group is known to be realised only finitely often, and among the groups compatible with the elementary constraints — at most four invariant factors, at most two of them divisible by any given odd prime, since J[p](ℚ) is isotropic for the Weil pairing when μ_p ⊄ ℚ — none is known not to be realised at all (Question 3 of the paper).

∞: 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), except for the groups [ ], [2] and [2,2,2,12], whose certificates were computed for this site (trivial_torsion_family.log, family_2_certificate.log, family_22212_certificate.log; the record's family for [2] is wrong, and for [2,2,2,12] the base is an elliptic curve of rank 1, where the same congruence argument applies directly). The literature source of a family is cited next to the record.
∞ ⊇: 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: the paper's Theorem 3.3 for [2,2,2,12], Leprévost's families with a class of order 13, 17, 19 and the [15] inherited from the [30] family on the simple side; 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). 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).

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.