Claude-Assisted Search Finds an Elliptic Curve of Rank at Least 30
Levent Alpöge and Ava Howell report a Claude-assisted elliptic-curve search producing a curve over the rationals with 30 independent rational points, giving an unconditional Mordell–Weil rank lower bound of 30 and a new computational record; exact rank 30 remains conditional on BSD and GRH, and the example does not resolve whether ranks over the rationals are uniformly bounded.
An August 2026 update to a MathOverflow discussion reports that Levent Alpöge and Ava Howell used Claude to find an elliptic curve over with Mordell–Weil rank at least 30. The curve was submitted to the ICARM Elliptic Curve Rank Leaderboard on August 20, 2026, extending the sequence of known rank records beyond the rank-29 example announced in 2024.
The public curve record supplies 30 rational points. The leaderboard states that each point is checked on the curve and that independence is certified by exact 2-descent, establishing the unconditional lower bound . The stronger statement that the rank is exactly 30 is conditional on BSD and GRH: the cited analytic calculation gives an upper bound of 31, and the root number forces even rank.
The visible sources identify the researchers as Alpöge and Howell and the assisting system only as Claude, without a model version or a detailed division of labor. A single high-rank example advances the computational record but does not settle whether Mordell–Weil ranks over are uniformly bounded or unbounded.