A Rank- $31$ Record for an Elliptic Curve over $\mathbb{Q}$

How large can the Mordell-Weil rank of an elliptic curve over $\mathbb{Q}$ be? Whether ranks are unbounded is open, and progress is measured by explicit records, tabulated by Dujella: rank $\ge 28$ from 2006, raised to $\ge 29$ by Elkies and Klagsbrun in 2024, and to $\ge 30$ three days before this one by the same team (see the related entry). Now $\ge 31$ , witnessed by an explicit curve $y^2 + xy + y = x^3 + x^2 + a_4 x + a_6$ with $a_4$ of 67 digits and $a_6$ of 99, carrying thirty-one independent rational points.

Result: Proved(see note)
Status: Partial result
AI contribution: AI co-developed
Method: Construction
Field: Elliptic curves
Posed by: Classical; rank records tabulated by Andrej Dujella
Year posed: —
Years open: —
Solved: 2026-08-23
Model: Claude
Vendor: Anthropic
Collaborators: Levent Alpöge, Ava Howell
Verification: Unreviewed
Publication: Announced
Significance: 50 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Two tiers, and only the first is the record. Rank $\ge 31$ is unconditional: 31 explicit points, independence asserted via the leaderboard's stated general practice of exact 2-descent (not reproduced here - see the verification note). Rank exactly 31 is conditional on GRH and BSD, per the submitters' commentary, in the same style as the sibling record's Bober-bound argument; no numeric derivation has been published for this curve specifically. The entry is a partial result because the open question - whether ranks are unbounded at all - remains unanswered by any single record.

What the AI did

The credit, in full, is the leaderboard's own commentary field on this entry: "BSD + GRH certified to rank 31, found by Claude, Levent Alpöge, and Ava Howell." No paper, no third-party comment of the kind the sibling record drew, no statement of division of labour, no account of what the model searched or proposed. Thinner disclosure than the sibling entry, which is already the weakest provenance in this catalog; classified the same regardless, since the evidence quality has not changed, only its brevity.

Verification

Recomputed by this site on 24 August 2026 from the leaderboard's own JSON, in exact rational arithmetic: all 31 witness points satisfy the curve equation with residual exactly zero (nine carry fractional coordinates, handled exactly rather than as floating point), all 31 are pairwise distinct, and the discriminant recomputed from the a-invariants matches the published value exactly. All 20 listed bad primes divide that discriminant and multiply out to account for the whole of it with nothing left over, and each passed a Miller-Rabin probable-primality check, including the 80-digit one. What was NOT checked, same limitation as the sibling entry: that the 31 points are independent in $E(\mathbb{Q})$ modulo torsion. The leaderboard states its site-wide practice is exact 2-descent with no floating point in the decision; that computation was not reproduced. Also unlike the sibling entry, no announcement article or public numeric derivation of the GRH+BSD argument (a Bober-bound $\Delta$ , a root number) exists for this specific curve at time of writing - the "exactly 31" claim rests on the submitters' commentary alone.

Sources

Related entries

ContinuesElliptic curve rank record

A Rank-313131 Record for an Elliptic Curve over Q\mathbb{Q}Q