Elliptic curves over ℚ of rank at least 30 and 31

The problem

The rational points on an elliptic curve over $\mathbb{Q}$ form a finitely generated group, and its rank measures how many independent points the curve has. Whether ranks can be arbitrarily large is a famous open question. The record of 29 was set by Elkies and Klagsbrun in 2024. Epoch AI's FrontierMath Open Problems asked for rank at least 30.

What AI did

On 20 August 2026 a curve credited to "Claude, with Levent Alpöge and Ava Howell" was posted to the Elliptic Curve Rank Leaderboard (curve #273), with 30 independent rational points and their regulator. On 23 August a curve with 31 independent points followed (curve #302).

How it was checked

A list of independent points can be checked directly, and the leaderboard page for the rank-31 curve says "Verified by ICARM on 2026-09-25". The original rank-30 submission silently dropped one point, apparently because of a parser bug that was later fixed.

Limitations

The points prove that the ranks are at least 30 and 31. That the ranks are exactly those numbers holds only assuming the Birch–Swinnerton-Dyer conjecture and the generalized Riemann hypothesis. Whether ranks are unbounded is still open, so we grade this a partial result. Problem description paraphrased from Epoch AI (CC BY 4.0).

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