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New Elliptic Curve Establishes a Record Rank Lower Bound of 30

A new elliptic curve has 30 certified independent rational points, surpassing the 2024 record. Its exact rank remains conditional.

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A newly submitted elliptic curve over the rational numbers has 30 independently certified rational points, establishing an unconditional rank lower bound of 30 and surpassing the previous record of at least 29.

The curve appeared on the NSF-backed ICARM Elliptic Curve Rank Leaderboard on August 20, 2026, under the pseudonymous account ranksunbounded. Its commentary subsequently credited Claude working with mathematicians Levent Alpöge and Ava Howell. Kevin Weil amplified the result on X the following day.

The mathematical record is stronger than the still-incomplete account of the AI’s role. ICARM publishes the curve, all 30 witness points, and an exact independence certificate. What has not yet appeared is a paper or technical report explaining how the candidates were generated, which model version conducted the search, or how responsibilities were divided between the humans and Claude.

1. The new curve has 30 certified independent points

Curve #273 is given in a global minimal Weierstrass model by

\[ y^2 + xy = x^3

  • 201769035260418549083594900060734240952308696994802735114305555x

+ 1151107939141058565733479426024323225135665982951300586808823640527729578307228357301072889377. \]

The leaderboard records a rank lower bound of 30, a naive height of 442.0854, a Faltings height of 34.7705, and a regulator of approximately \(1.0721 \times 10^{37}\). It also supplies the curve’s conductor, discriminant, primes of bad reduction, and the complete coordinates of 30 rational points.

Those points are not merely 30 examples selected from an infinite set. For an elliptic curve \(E\) over the rationals, the Mordell–Weil theorem describes the rational points as a finitely generated abelian group:

\[ E(\mathbb{Q}) \cong E(\mathbb{Q})_{\mathrm{tors}} \oplus \mathbb{Z}^{r}. \]

The integer \(r\) is the rank. Proving that 30 rational points are independent modulo torsion proves that the free part contains a copy of \(\mathbb{Z}^{30}\), and therefore that \(r \geq 30\).

ICARM says its verifier first checks that every submitted point lies on the stated curve. It then certifies independence through an exact 2-descent computation using quadratic characters at good primes, following methods associated with John Cremona and Armand Brumer. The acceptance decision does not depend on floating-point approximations.

The submission briefly exposed an infrastructure problem. According to the curve commentary, the original form silently dropped one witness point, possibly because of a parser bug. A linked code change made the parser reject malformed point lines instead of discarding extra coordinates or mishandling line separators. The corrected record now displays all 30 witnesses. That parser fix credits Claude Fable 5 as a code co-author, although the commit alone does not establish which Claude model performed the mathematical search.

2. “Rank at least 30” is the unconditional result

The strongest unconditional statement currently supported by the public evidence is that the curve’s rank is at least 30. The 30 exact points and their independence certificate supply a lower bound; they do not, by themselves, exclude the existence of a 31st or 32nd independent generator.

The leaderboard commentary reports a conditional argument for exact rank 30. Applying Jonathan Bober’s explicit-formula method with parameter \(\Delta=4.25\) gives an upper bound of 31 for the analytic rank, assuming the relevant generalized Riemann hypothesis. The curve has global root number \(+1\), forcing the analytic rank to have even parity. Under the Birch and Swinnerton–Dyer conjecture, which identifies the algebraic rank with the analytic rank, the known lower bound and conditional upper bound meet at 30.

This means “rank 30” is a reasonable conditional description, but “rank at least 30” is the verified, assumption-free result. The discovery does not prove the generalized Riemann hypothesis or the Birch and Swinnerton–Dyer conjecture.

The distinction is familiar in high-rank searches. Computing an upper bound can be much harder than finding many independent points, particularly when the curve has enormous coefficients and conductor. Bober’s method was developed precisely to obtain conditional analytic-rank bounds without directly evaluating an impractically large number of terms in the curve’s \(L\)-function.

3. The previous record was two years old

Noam Elkies and Zev Klagsbrun announced the previous record in August 2024: an elliptic curve with 29 independent rational points. Their result raised a record that had remained at 28 since Elkies’ 2006 construction.

The 2024 curve emerged from a sieve search through specializations of a rank-17 fibration on a K3 surface. According to Elkies’ announcement, Klagsbrun found 12 additional independent points beyond the 17 supplied by the underlying family. Conditional arguments then showed that the rank was exactly 29.

That improvement required a dramatic increase in search scale. Quanta reported that Elkies had examined millions of curves for the 2006 record, while Klagsbrun’s later computational method could sift through tens of trillions. Despite that expansion, the record moved by only one.

Rank 30 had consequently become a compact test of research-level mathematical search. Epoch AI’s FrontierMath Open Problems page specified the target as a Weierstrass equation over \(\mathbb{Q}\) accompanied by exactly 30 independent rational points. The verifier was designed to accept exact integer or fractional coordinates rather than numerical approximations.

Curve #273 supplies the required mathematical object. Whether it resulted directly from that benchmark environment has not been documented publicly, so the match between the published task and the submission should not be treated as proof of provenance.

4. Claude is credited, but the discovery process is not yet documented

The curve was initially presented under a pseudonym. The leaderboard’s editable commentary later stated that the result came from Claude with Alpöge and Howell. Weil’s X post framed it as another example of humans and AI learning new things together.

That attribution supports calling the result AI-assisted, but it does not yet answer the central research questions. There is no public description of the search family, candidate-ranking method, software stack, prompt history, compute budget, number of rejected candidates, or the point-search and saturation procedures used before submission.

The available evidence also does not securely identify the model version used for the mathematical work. The related parser commit names Claude Fable 5, and Anthropic describes Fable 5 as its generally available Mythos-class model for long-running analytical and scientific tasks. Still, code authorship on the leaderboard is not proof that Fable 5 generated the curve.

A formal account could distinguish several materially different forms of AI contribution: proposing a new mathematical family, translating a human strategy into optimized code, autonomously running and adapting a search, locating additional rational points, or preparing and checking the final certificate. Until such an account appears, the result should be described as credited to a human–Claude collaboration rather than as an autonomous AI discovery.

5. Why the result is useful evidence for AI-assisted research

High-rank elliptic-curve searches have an unusually favorable verification structure. Discovery is difficult, but a successful output can be expressed as a finite collection of exact integers and rational numbers. Independent software can then check the equation, every point, and the required linear independence without trusting the model’s explanation.

That separates two questions often conflated in AI research announcements: whether the submitted mathematical object is valid, and whether the reported account of how the AI found it is complete. For curve #273, the first question has unusually strong public evidence. The second remains open pending a reproducible technical report.

The record does not determine whether elliptic-curve ranks over \(\mathbb{Q}\) are unbounded. It supplies one new extreme example, raising the known lower-bound record by one. Its immediate practical significance is therefore in arithmetic geometry and in the evaluation of human–AI mathematical search, not in deployed elliptic-curve cryptography.

Frequently Asked Questions

Is the curve’s rank definitely 30?

It is unconditionally known to have rank at least 30. Exact rank 30 currently depends on GRH and the Birch and Swinnerton–Dyer conjecture.

Why are 30 rational points enough to set the record?

The verifier proves that the points are independent modulo torsion. Their span therefore forms a rank-30 subgroup of the curve’s rational points.

Did Claude discover the curve autonomously?

The leaderboard credits Claude with Levent Alpöge and Ava Howell, but no public report yet explains the division of work. Autonomous discovery has not been established.

Was Claude Fable 5 the model used?

A related parser-fix commit credits Claude Fable 5, but the mathematical search’s model version has not been publicly confirmed.

Does the result affect elliptic-curve cryptography?

Not directly. The record concerns rational points on a curve over \(\mathbb{Q}\), whereas deployed elliptic-curve cryptography generally uses groups over finite fields.

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