Elliptic Curves over $\mathbb{Q}$ of Large Rank

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

An elliptic curve E over the rationals can be written in the general Weierstrass form

    E : y^2 + a1 x y + a3 y = x^3 + a2 x^2 + a4 x + a6,   a1, a2, a3, a4, a6 in Z.

By the Mordell-Weil theorem its group of rational points is finitely generated,

    E(Q) ≅ E(Q)_tors ⊕ Z^r,

and the non-negative integer r is the rank of E.

## Task

Find an elliptic curve E over Q of rank at least 30, and exhibit 30 rational
points on E that are linearly independent in E(Q) (equivalently, independent modulo
torsion). These 30 independent points force rank E(Q) >= 30.

## 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 two keys, "curve" and
"x_coords":

  - "curve": the five coefficients [a1, a2, a3, a4, a6] of the Weierstrass model above,
    each an integer written as a string (e.g. ["1", "0", "0", "-7", "12"]). The
    coefficients must define a non-singular curve (nonzero discriminant).

  - "x_coords": a list of exactly 30 x-coordinates, each the x-coordinate of a
    rational point on E. Give each as a string that is an exact integer (e.g. "5") or
    fraction (e.g. "-3/2"). You only supply x-coordinates: the verifier reconstructs a
    corresponding y (the two choices are the inverse points \(P\) and \(-P\); choosing either
    does not affect independence). Every x MUST be the x-coordinate of a genuinely rational
    point -- the reconstructed y must be rational -- otherwise the point is not in E(Q) and
    the submission is rejected.

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

{"curve": ["0", "0", "0", "0", "1"], "x_coords": ["2", "0", "-1"]}

Approximate or floating-point coordinate values will not be accepted.

Epoch marks this problem solved (human + AI) on 2026-08-20. This is Epoch’s classification; the solution has not been independently checked for this record.

Epoch solution update

First solved with: Claude (internal Anthropic model)

An elliptic curve over $\mathbb{Q}$ with rank at least 30 was posted to the Elliptic Curve Rank Leaderboard on August 20. The solution is credited to Claude, with Levent Alpöge and Ava Howell, so we are marking the problem as solved by AI. On August 23, a curve of rank at least 31 was posted and credited to the same team. Assuming BSD and GRH, these curves have rank exactly 30 and 31, respectively.