Claude-assisted search breaks the elliptic curve rank record: rank 30, then 31
An elliptic curve over Q with rank at least 30 was reported on 20 Aug 2026 and one with rank ≥31 on 23 Aug. These broke the Elkies–Klagsbrun rank-29 record from 2024. The rank-31 curve has 31 explicit independent rational points, so the bound is unconditional. The ICARM record page credits Claude with L. Alpöge and A. Howell.
Key facts
- Previous record: rank ≥ 29 (Elkies–Klagsbrun, 2024); earlier ≥ 28 (Elkies, 2006)
- Rank 30 on 20 Aug; rank 31 on 23 Aug 2026; first submitted under the name 'ranksunbounded'
- 31 independent rational points given explicitly
Science result
- Field
- mathematics / arithmetic geometry
- Problem
- How large can the rank of an elliptic curve over Q be?
- Result
- Explicit elliptic curve with rank at least 31, a new record.
- AI system
- Claude
- Human role
- AI-assisted search with human researchers (Alpöge, Howell)
- Verification
- Independent points checkable by computer; listed by the rank record tables
- Status
- confirmed
What happened
AI-directed searches through families of elliptic curves found new record-rank examples twice in one week.
Why it matters
Rank records move very rarely (2006, 2024). Two in a week signal AI's strength at large, structured searches in number theory.
Changelog
- 2026-09-29: created
Related events
- Claude-assisted constructions complete Hadamard matrices for every order below 2000, including 668 ★★★
Sources (3)
- officialICARM: new record-breaking elliptic curve reported
- docsAndrej Dujella: history of elliptic curve rank records
- officialEpoch AI open problems: elliptic curve rank
id: 2026-08-23-elliptic-curve-rank-record-31 · updated 2026-09-29 · open in the interactive timeline