Mathematics / Milestone

Elliptic-Curve Rank Records

How high can the rank of an elliptic curve over the rationals go?

The source trail 6

Findings, with context

DiscussionAug 24, 2026

A source discussing Elliptic-curve ranks

tsujimotter.hatenablog.com contains the excerpt below in a discussion related to Elliptic-curve ranks. Read the source for the surrounding context.

“どうやら今回の発見には、数学者の Levent Alpöge、Ava Howell、そして生成AIの Claude が関わっていたようです。”

This excerpt is evidence of what the linked page says, not verification of its claims. Source reports an update on 2026-09-04; this is not necessarily the original posting date.

Source pageIndirect account · Found Sep 20

tsujimotter.hatenablog.com

DiscussionFound Sep 20, 2026

A source discussing Elliptic-curve ranks

mathworld.wolfram.com contains the excerpt below in a discussion related to Elliptic-curve ranks. Read the source for the surrounding context.

“Thirty-one linearly independent rational points prove unconditionally that its rank is at least 31.”

This excerpt is evidence of what the linked page says, not verification of its claims. Publication date has not been established.

Source pageIndirect account · Found Sep 20

mathworld.wolfram.com

DiscussionAug 27, 2026

A source discussing Elliptic-curve ranks

groundtruth.day contains the excerpt below in a discussion related to Elliptic-curve ranks. Read the source for the surrounding context.

“Curve #273 was submitted on August 20, 2026 with rank at least 30; curve #302 followed on August 23 with rank at least 31.”

This excerpt is evidence of what the linked page says, not verification of its claims. Source reports an update on 2026-08-27; this is not necessarily the original posting date.

Source pageIndirect account · Found Sep 20

groundtruth.day

DiscussionFound Sep 20, 2026

A source discussing Elliptic-curve ranks

dujella.github.io contains the excerpt below in a discussion related to Elliptic-curve ranks. Read the source for the surrounding context.

“The highest rank of an elliptic curve which is (unconditionally) known exactly (not only a lower bound for rank) is equal to 20”

This excerpt is evidence of what the linked page says, not verification of its claims. Publication date has not been established.

Source pageIndirect account · Found Sep 20

dujella.github.io

AnnouncementAug 23, 2026

Milestone completed

Milestone completed: an example with rank at least 31. This watch remains active for higher records and further research.

Elliptic-curve rank record reaches at least 31

ICARM records a curve with 31 independent rational points, credited to Claude, Levent Alpöge, and Ava Howell. Epoch identifies the related challenge as solved by human + AI. The lower bound is at least 31; the reported exact rank assumes BSD and GRH.

“BSD + GRH certified to rank 31, found by Claude, Levent Alpöge, and Ava Howell.”

August 23 is the original submission date. The leaderboard supplies 31 independent points and certifies the lower bound; its commentary makes exact rank 31 conditional. This does not prove BSD or show that elliptic-curve ranks are unbounded. ProofWatch has not independently rerun the arithmetic verification.

Ava Howell · credits Claude and Levent AlpögeOriginal source · Found Sep 20

ICARM rank leaderboard

Earlier records & findings

AnnouncementAug 20, 2026

Superseded

The rank-at-least-30 target was reached on August 20. Its record was surpassed by the rank-at-least-31 example on August 23; the construction remains part of the history.

The rank-at-least-31 record

A rank-at-least-30 curve passes the earlier record

ICARM's August 20 entry supplies 30 independent points. Its commentary credits Claude, Levent Alpöge, and Ava Howell, and Epoch treats the finite rank-at-least-30 challenge as solved by human + AI. Three days later, the same team posted the stronger lower bound of 31.

The page records a corrected witness-point submission after a parser issue; its current entry lists 30 independent points. Exact rank 30 is conditional on BSD and GRH. Superseded means a higher record followed, not that this result was retracted or that the broader rank question was solved.

ICARM entry · credits Claude, Levent Alpöge and Ava HowellOriginal source · Found Sep 20

ICARM rank leaderboard