Number theory·elliptic:rank-record
Largest known rank of an elliptic curve over the rationals
No independent check recorded yet. A formal artifact, declaration or published object is attached, but no rebuild of it is recorded here.
Fidelity F0: Absent. No formal statement is attached to the result.
The rank of an elliptic curve over the rational numbers counts its independent rational points of infinite order. Whether ranks are unbounded is open; the record is a lower bound, shown by exhibiting independent points.
Record history
20 steps·rank (lower bound), higher is betterRecords rose from 3 in 1938 to 28 when Noam Elkies found his curve in 2006, and stood there for eighteen years until Elkies and Zev Klagsbrun reached 29 in 2024. In August 2026 curves of rank at least 30 and 31 were submitted to ICARM's Elliptic Curve Rank Leaderboard, which verifies each submission before admitting it; the rank-31 entry is credited to Claude, Levent Alpöge and Ava Howell. Steps before 2026 follow Andrej Dujella's table of rank records, which cites each original paper.
Curve 302. The leaderboard's entry reads: found by Claude, Levent Alpöge, and Ava Howell. Rank exactly 31 is certified under BSD and GRH; the unconditional result is the lower bound shown by 31 independent points.
Curve 273, submitted to ICARM's leaderboard by the user ranksunbounded. Dujella's table credits Alpöge and Howell; Scientific American reports that they used an internal variant of Claude.
From Dujella's table of rank records.
From Dujella's table of rank records; the record for eighteen years.
From Dujella's table of rank records.
From Dujella's table of rank records.
From Dujella's table of rank records.
From Dujella's table of rank records.
From Dujella's table of rank records.
From Dujella's table of rank records.
From Dujella's table of rank records.
From Dujella's table of rank records.
From Dujella's table of rank records.
From Dujella's table of rank records.
From Dujella's table of rank records.
From Dujella's table of rank records.
From Dujella's table of rank records.
From Dujella's table of rank records.
From Dujella's table of rank records.
From Dujella's table of rank records.
AI activity
How grades workAn elliptic curve over the rationals of rank at least 30, the first above Elkies and Klagsbrun's 29.
Reasoning and sources
Autonomy
Reported as found by two mathematicians using an internal variant of Claude; the leaderboard's entry for this curve names only its submitter.
An elliptic curve over the rationals of rank at least 31, verified by ICARM's leaderboard.
Reasoning and sources
Autonomy
The leaderboard credits the curve jointly to Claude and two mathematicians.
Fidelity
How fidelity is gradedF0 no formal statement. Absent. No formal statement is attached to the result.
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qed.bot, “Largest known rank of an elliptic curve over the rationals”, https://qed.bot/t/elliptic-rank-record, as of 30 Sep 2026.