Record Rank for an Elliptic Curve over $\mathbb{Q}$
partialconfidence 70%
VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
Precise statement
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. Now $\ge 30$, witnessed by an explicit curve $y^2 + xy = x^3 + a_4 x + a_6$ with $a_4$ of 63 digits and $a_6$ of 94, carrying thirty independent rational points.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement
fidelity, correctness, priority, or novelty.
What AI did
Claude
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.
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 $E(\mathbb{Q})$ 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.
Two tiers, and only the first is the record. Rank $\ge 30$ is unconditional, being thirty explicit independent points. Rank exactly 30 is conditional: applying Bober's bound (arXiv:1112.1503) with $\Delta = 4.25$ gives an analytic rank of at most 31, and the root number is $+1$ 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. Superseded three days later by this project's own rank $\ge 31$ record (see the related entry); left unedited otherwise as a record of what was known at the time.
Known method families
construction (source-reported)
Source-reported tools: construction.
Independent: unknown · difference confidence: 0
What remains uncertain
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.
VibeMath has not independently verified the mathematical claim.
AI-attempt independence and training-data exposure are unknown unless explicitly documented.
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.