A Genus 2 Curve over the Rationals with a Rational Torsion Point of Prime Order at Least 31

FrontierMath: Open Problems lists this problem with notability “Moderately interesting”, field “Arithmetic geometry” and type “Construction - Finite; Record Breaking”. Each problem has a bespoke verifier program, so a proposed answer can be checked automatically.

Verifiable task — Epoch prompt, verbatim

For each squarefree polynomial f(x) in Z[x] of degree 6, the curve

    C_f : y^2 = f(x)

is hyperelliptic of genus 2, and its Jacobian J_f = Jac(C_f) is an abelian surface over Q.

A rational point of J_f can be given in Mumford coordinates by a pair of polynomials (u, v)
over Q with

  - u monic of degree 2,
  - deg v <= 1,
  - u dividing v^2 - f.

Such a pair describes the effective divisor D of degree 2 cut out by u(x) = 0 and y = v(x),
and the corresponding point of J_f is the divisor class [D - K], where K is the canonical
class of C_f: the degree-2 divisor cut out by x = constant, equivalently the sum of the two
points at infinity of the smooth projective model. (K is a rational divisor class even when
the two points at infinity are individually irrational, so [D - K] is a rational point of J_f
for every such pair (u, v).)

# Task

Find a squarefree polynomial f(x) in Z[x] of degree exactly 6, together with a point of
J_f(Q) whose order is a prime number p >= 31.

(There is no loss of generality in taking f to have integer coefficients: f and c^2 * f define
isomorphic curves for any c in Q*, so denominators can be cleared.)

# Submission

Write your answer to a file, then submit the path to that file using the `submit_answer` tool.
The file must contain a single JSON object with exactly three keys, "coeffs", "u", and "v",
each a list of coefficients given as strings, CONSTANT TERM FIRST:

  - "coeffs": the 7 coefficients of f, i.e. ["c0", "c1", ..., "c6"] meaning
    f(x) = c0 + c1 x + ... + c6 x^6. Each must be an integer, and c6 must be nonzero so that f
    has degree exactly 6.
  - "u": the 3 coefficients of u, i.e. ["u0", "u1", "u2"] meaning u(x) = u0 + u1 x + u2 x^2.
  - "v": the 2 coefficients of v, i.e. ["v0", "v1"] meaning v(x) = v0 + v1 x.

Every entry must be a string holding an exact integer such as "5" or an exact fraction such as
"-3/2". Everything is checked with exact arithmetic, so approximate or floating-point values will
not be accepted.

The file should contain exactly this JSON object and nothing else. The example below only
illustrates the format; it is not a valid answer:

{"coeffs": ["1", "0", "1", "0", "1", "0", "1"], "u": ["1", "0", "1"], "v": ["0", "0"]}

Epoch marks this problem solved (AI) on 2026-07-30. This is Epoch’s classification; the solution has not been independently checked for this record.

Epoch solution update

First solved by: GPT-5.6 Sol

In a pre-release test of FM:OP done by Epoch on July 30, 2026, GPT-5.6 Sol found a solution to this problem and successfully passed the problem verifier. Sol’s solution is a curve from a public repository which Sol found through searching. Therefore, we are counting this problem as solved on release. The repository is maintained by Edgar Costa and contains the work of Edgar Costa, Noam Elkies, Sachi Hashimoto, Aashraya Jha, Kimball Martin, Bjorn Poonen, and John Voight. The curve also appears in a public repository maintained by Timo Keller.

In this instance, the solution to the problem was latent in the existing literature: the curve, which is part of a substantial project to create a dataset of curves, is in a public database, and would have been recognized as a solution had someone checked its torsion subgroup. Moving forward, we will ask for a rational point of order 37, which is the next target. Ideally, the new target is beyond the current research frontier, so finding a solution there will hopefully require a new idea.