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Largest known rank of an elliptic curve over Q

The largest rr for which an elliptic curve E/QE/\mathbb{Q} with rank⁡E(Q)≥r\operatorname{rank} E(\mathbb{Q}) \ge r is known, with rr independent points exhibited.

Current best · higher is better
≥31\ge 31
Claude with Alpöge and Howell, 23 Aug 2026 · entry
steps
14
by AI
2
since
1982
1990200020102020152025301982: ≥ 12 (Mestre)1986: ≥ 14 (Mestre)1992: ≥ 15 (Mestre)1992: ≥ 17 (Nagao)1992: ≥ 19 (Fermigier)1993: ≥ 20 (Nagao)1994: ≥ 21 (Nagao and Kouya)1997: ≥ 22 (Fermigier)1998: ≥ 23 (Martin and McMillen)2000: ≥ 24 (Martin and McMillen)2006: ≥ 28 (Elkies)2024: ≥ 29 (Elkies and Klagsbrun)2026-08-20: ≥ 30 (Claude with Alpöge and Howell)2026-08-23: ≥ 31 (Claude with Alpöge and Howell)

The line is the frontier over time. Filled dots are steps that moved it; muted dots are results that did not. Orange dots are catalog entries, results with AI in the loop. Hollow dots are candidates under review and never move the line. Grey dots along the bottom edge are results from before the quantity had a number, placed there because they have no value on this axis. Dots that would overlap are nudged sideways a few pixels. Hover a dot for its value and attribution.

About this frontier

Whether ranks are bounded is open and experts disagree. The record is a construction race: each row is an explicit curve with that many independent rational points, unconditionally. Exact ranks (rather than lower bounds) usually need GRH and are not what this record tracks. It moved about once a decade until 2024, then twice in three days in August 2026.

Every step, newest first

DateValueWhoModelStatusSource
23 Aug 2026≥31\ge 31bestClaude with Alpöge and HowellClaudeAI step
unreviewed
entry
20 Aug 2026≥30\ge 30Claude with Alpöge and HowellClaudeAI step
unreviewed
entry
2024≥29\ge 29Elkies and Klagsbrun–historicalsource ↗
2006≥28\ge 28
Stood for eighteen years.
Elkies–historicalsource ↗
2000≥24\ge 24Martin and McMillen–historicalsource ↗
1998≥23\ge 23Martin and McMillen–historicalsource ↗
1997≥22\ge 22Fermigier–historicalsource ↗
1994≥21\ge 21Nagao and Kouya–historicalsource ↗
1993≥20\ge 20Nagao–historicalsource ↗
1992≥19\ge 19Fermigier–historicalsource ↗
1992≥17\ge 17Nagao–historicalsource ↗
1992≥15\ge 15Mestre–historicalsource ↗
1986≥14\ge 14Mestre–historicalsource ↗
1982≥12\ge 12Mestre–historicalsource ↗

Historical rows are Dujella's History of elliptic curves rank records, the reference table the field itself maintains; years and attributions as given there. Only rows that raised the record are included.

Changelog1 change
  • Curatoradded this entry

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