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

How large can the Mordell-Weil rank of an elliptic curve over Q\mathbb{Q} be? Whether ranks are unbounded is open, and progress is measured by explicit records, tabulated by Dujella: rank ≥28\ge 28 from 2006, raised to ≥29\ge 29 by Elkies and Klagsbrun in 2024. Now ≥30\ge 30, witnessed by an explicit curve y2+xy=x3+a4x+a6y^2 + xy = x^3 + a_4 x + a_6 with a4a_4 of 63 digits and a6a_6 of 94, carrying thirty 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
2026-08-20
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 ≥30\ge 30 is unconditional, being thirty explicit independent points. Rank exactly 30 is conditional: applying Bober's bound (arXiv:1112.1503) with Δ=4.25\Delta = 4.25 gives an analytic rank of at most 31, and the root number is +1+1 so the rank is even, hence 30 - but that argument assumes GRH, and equating analytic rank with rank assumes BSD. The entry is a partial result because the open question is whether ranks are unbounded at all, which no single record answers.

What the AI did

The credit, in full, is a comment on the leaderboard entry: "it was Claude, with Levent Alpöge and Ava Howell!" Bartosz Naskręcki, who works on these curves, congratulated "Ava Howell, Levent Alpöge and the team Anthropic" publicly. That is the whole of the disclosure: no paper, no statement of division of labour, and no account of what the model searched or proposed. The tier below is inferred from the wording rather than read off an author's description, which is weaker evidence than every arXiv entry in this catalog.

Verification

Recomputed by this site on 21 August 2026 from the leaderboard's own JSON, in exact rational arithmetic: all thirty witness points satisfy the curve equation with residual exactly zero, all thirty are distinct, nineteen are integral, and the discriminant recomputed from the a-invariants matches the published value, with all fourteen listed bad primes dividing it and together factoring it completely. What was NOT checked here is the one thing the record actually asserts - that the thirty points are independent in E(Q)E(\mathbb{Q}) modulo torsion. The leaderboard states it certifies independence by exact 2-descent with no floating point in the decision; that computation was not reproduced. The page is also living data: its commentary records that the original submission silently dropped a witness point through a parser bug, corrected three hours later.

Sources

Changelog1 change

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