A genus-2 curve with a rational torsion point of prime order at least 31

GPT-5.6 Sol passed Epoch AI's verifier with such a curve, but the curve came from an existing public database; Epoch itself says the solution was latent in the literature.

Problem
Is there a genus-2 curve over Q\mathbb{Q} whose Jacobian has a rational torsion point of prime order at least 31?
Credit
GPT-5.6 Sol (OpenAI) · Retrieved from existing data
Checked
Passed Epoch AI's verifier
Timeline
Previous largest known prime order 29 (Leprévost) → 2026-07-30
Sources
Epoch AI problem page
PRIOR
ARTAlready in a public database

The problem

For a genus-2 curve over Q\mathbb{Q}, the rational points of finite order on its Jacobian form a finite group. Which prime orders can occur is not known; the largest previously known was 29, due to Leprévost. Epoch AI's FrontierMath Open Problems asked for a curve with a rational torsion point of prime order at least 31.

What AI did

On 30 July 2026, OpenAI's GPT-5.6 Sol submitted a curve that passed Epoch AI's verifier. The curve, however, was already in a public database: Edgar Costa's ModularAbelianSurfaces repository, which collects work by Costa, Elkies, Hashimoto, Jha, Martin, Poonen and Voight. It also appears in a repository maintained by Timo Keller.

How it was checked

The verifier confirmed the curve and the torsion point; such a claim can be checked by direct computation.

Limitations

Epoch AI itself says the solution "was latent in the existing literature" and has raised the target to order 37. The model found a known answer rather than a new one, so we grade this "prior". Problem description paraphrased from Epoch AI (CC BY 4.0).

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