Elliptic curves over the rationals of rank at least 30 and at least 31
Two explicit elliptic curves over the rationals carry 30 and 31 independent rational points, raising the Mordell-Weil rank record twice in four days.
- Lab
- Independent
- Model
- Claude
- Field
- Mathematics
- Date
- 2026-08-23
- Human collaborators
- Levent Alpöge, Ava Howell
Sources
Original work
Independent commentary
What was found
Whether the Mordell-Weil rank of an elliptic curve over the rationals is bounded is open, and progress is measured by explicit records: rank at least 28 from 2006, raised to at least 29 by Elkies and Klagsbrun in 2024. Curve 273 on the elliptic-rank leaderboard reaches rank at least 30 on 2026-08-20 and curve 302 reaches at least 31 on 2026-08-23, each witnessed by explicit independent points. The method is the human Elkies-Klagsbrun elliptic-K3 family search, with the model assisting inside it. Both entries are recorded as two tiers: the lower bound is what the explicit points establish, while exact rank is claimed only conditionally on the Birch-Swinnerton-Dyer conjecture and the generalized Riemann hypothesis.
Novelty check
The rank record ladder is well documented and tabulated by Dujella, which makes the prior state of the art unambiguous: Elkies rank at least 28 in 2006 and Elkies and Klagsbrun rank at least 29 in 2024. Both new curves are given explicitly with their a-invariants, conductor, discriminant and regulator, so the claim is checkable against that table rather than resting on an assertion. No intervening record appears. These are new curves, not a rediscovery.
Caveats and known objections
Recorded only as leaderboard database entries: there is no preprint, no transcript and no refereeing, and nothing is formalized. The two records are kept in one entry because they are the same method and the same people four days apart. Independence of the points is asserted through the leaderboard's stated general practice of exact 2-descent rather than reproduced on the page, and the exact-rank claim is conditional on BSD and GRH with no numeric derivation published for either curve. What the model contributed is documented only by the leaderboard's own commentary field, which reads in full: 'BSD + GRH certified to rank 31, found by Claude, Levent Alpöge, and Ava Howell.' Autonomy is graded ai-assisted on that thin evidence, because the search framework is the humans' and no stronger reading is defensible. Neither record answers the open question of whether ranks are unbounded.
Also recorded at
vibemathedelliptic-curve-rank-record-thirty-one
Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.
vibemathedelliptic-curve-rank-record-thirty
Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.
Nobody outside the lab has checked this yet.
Reading the primary source closely enough to say whether it supports the claim counts as a check, and you are credited on the entry.
Or on GitHub: submit a check challenge the grade send a correction or send a pull request
Flag this for triage
Signals order the review queue and nothing else. They are never published, and they never move a grade: that takes a citation.
Entry history (1 event)
- AddedEntered the registry graded Claimed and AI-assisted.
Entries are never deleted. A grade that does not hold up is downgraded on the record, with the reason beside it.
Graded claimed for verification and ai-assisted for autonomy. What these mean.
Cite this entry
whataifound.org. (2026). Elliptic curves over the rationals of rank at least 30 and at least 31. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-23-elliptic-rank-31BibTeX
@misc{whataifound-independent-2026-31,
title = {Elliptic curves over the rationals of rank at least 30 and at least 31},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Independent. Verification: Claimed. Autonomy: AI-assisted.},
url = {https://whataifound.org/finding/2026-08-23-elliptic-rank-31}
}