# Largest known rank of an elliptic curve over the rationals

A record target in qed.bot, the register of claims of AI work in mathematics. The register grades the evidence attached to each claim; it never rules on whether a proof is correct.

- Identifier: elliptic:rank-record
- Collection: Number theory
- Page: https://qed.bot/t/elliptic-rank-record
- Record: at least 31, held by Claude, Levent Alpöge and Ava Howell, since 23 Aug 2026
- Machine-checked: no independent check has verified it
- Fidelity F0: Absent. No formal statement is attached to the result.
- Source: https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html

## Record history

- 1938: at least 3, G. Billing; moved the record. https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html
- 1945: at least 4, A. Wiman; moved the record. https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html
- 1974: at least 6, David Penney and Carl Pomerance; moved the record. https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html
- 1975: at least 7, David Penney and Carl Pomerance; moved the record. https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html
- 1977: at least 8, Fritz Grunewald and Rainer Zimmert; moved the record. https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html
- 1977: at least 9, Armand Brumer and Kenneth Kramer; moved the record. https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html
- 1982: at least 12, Jean-François Mestre; moved the record. https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html
- 1986: at least 14, Jean-François Mestre; moved the record. https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html
- 1992: at least 15, Jean-François Mestre; moved the record. https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html
- 1992: at least 17, Kazuhiko Nagao; moved the record. https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html
- 1992: at least 19, Stéphane Fermigier; moved the record. https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html
- 1993: at least 20, Kazuhiko Nagao; moved the record. https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html
- 1994: at least 21, Kazuhiko Nagao and Tomonori Kouya; moved the record. https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html
- 1997: at least 22, Stéphane Fermigier; moved the record. https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html
- 1998: at least 23, Robert Martin and William McMillen; moved the record. https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html
- 2000: at least 24, Robert Martin and William McMillen; moved the record. https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html
- 2006: at least 28, Noam Elkies; moved the record. https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html
- 2024: at least 29, Noam Elkies and Zev Klagsbrun; moved the record. https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html
- 20 Aug 2026: at least 30, Levent Alpöge and Ava Howell, with Claude, with AI; moved the record. https://icarm.io/news/new-record-breaking-elliptic-curve-reported/
- 23 Aug 2026: at least 31, Claude, Levent Alpöge and Ava Howell, with AI; moved the record. https://elliptic-rank.icarm.cloud/curve/302

## AI claims

### Claude, 20 Aug 2026

An elliptic curve over the rationals of rank at least 30, the first above Elkies and Klagsbrun's 29.

- Outcome: superseded
- Autonomy A1: Collaborative. A human and a system worked the problem together, or the system played a supporting role.
- Evidence V2: Artifact reported. A proof or a constructed object is published, but nobody independent of its authors has rebuilt it.
- Fidelity F0: Absent. No formal statement is attached to the result.
- With: Levent Alpöge, Ava Howell
- Source: ICARM, New record breaking elliptic curve reported, https://icarm.io/news/new-record-breaking-elliptic-curve-reported/
- Source: Scientific American, Anthropic's AI steals mathematicians' record for most complicated curve, https://www.scientificamerican.com/article/anthropics-ai-steals-mathematicians-record-for-most-complicated-curve/

### Claude, 23 Aug 2026

An elliptic curve over the rationals of rank at least 31, verified by ICARM's leaderboard.

- Outcome: record
- Autonomy A1: Collaborative. A human and a system worked the problem together, or the system played a supporting role.
- Evidence V2: Artifact reported. A proof or a constructed object is published, but nobody independent of its authors has rebuilt it.
- Fidelity F0: Absent. No formal statement is attached to the result.
- With: Levent Alpöge, Ava Howell
- Source: ICARM Elliptic Curve Rank Leaderboard, curve 302, https://elliptic-rank.icarm.cloud/curve/302

Cite as: qed.bot, "Largest known rank of an elliptic curve over the rationals", https://qed.bot/t/elliptic-rank-record, as of 30 Sep 2026. The register's data is published under CC BY 4.0: https://creativecommons.org/licenses/by/4.0/