---
id: "2026-08-23-elliptic-curve-rank-record-31"
url: "https://postcutoff.com/e/2026-08-23-elliptic-curve-rank-record-31/"
as_of: "2026-11-13T13:49:00+01:00"
date: "2026-08-23"
date_precision: day
category: science
importance: 3
confidence: medium
status: [Result confirmed]
verification: Expert-checked
sources: 3
editor: human editor
human_review: null
version: "2026-10-04"
---

As of: 2026-11-13 13:49 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-08-23-elliptic-curve-rank-record-31/

# Claude-assisted search breaks the elliptic curve rank record

Full title: 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

## 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.

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 115 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 54 days after its cutoff
- Gemini 4 Hydrogen (training cutoff March 2026): 145 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 84 days after its cutoff

## Sources

1. [ICARM: new record-breaking elliptic curve reported](https://icarm.io/news/new-record-breaking-elliptic-curve-reported/) (icarm.io, official)
2. [Andrej Dujella: history of elliptic curve rank records](https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html) (web.math.pmf.unizg.hr, docs)
3. [Epoch AI open problems: elliptic curve rank](https://epoch.ai/frontiermath/open-problems/elliptic-curve-rank) (epoch.ai, official)

## Changes

- 2026-09-29 (filed): Created

## Related

- 2026-08-12: [Claude-assisted constructions complete Hadamard matrices for every order below 2000](https://postcutoff.com/e/2026-08-12-claude-hadamard-matrices-below-2000/index.md)
- People: [Levent Alpöge](https://postcutoff.com/person/levent-alpoge/)