AI system Claude helped find an elliptic curve with 30 independent points; the previous record was 29

AI system Claude helped find an elliptic curve with 30 independent points; the previous record was 29

On August 20, the ICARM online registry of elliptic curves added entry No. 273. This curve has 30 rational points, and the lower bound on its rank is 30. In the entry’s comment, its author credited Claude, Levent Alpöge, and Ava Howell with the discovery. The previous record-holding curve, found in 2024, had 29 independent points.

In this problem, an elliptic curve is a mathematical equation whose solutions can be added according to a specific rule. Rational points are solutions whose coordinates are fractions. A point is independent if it cannot be obtained by adding other points, and each independent point contributes a new direction to the set of solutions. The number of these directions is called the rank.

The page for entry No. 273 provides the curve’s equation and the coordinates of all 30 points. ICARM verifies that every point lies on the curve and then confirms their independence using an exact 2-descent, a calculation that does not rely on numerical approximations.

The previous record from 2024 concerned a different curve with 29 independent points. The new entry contains a new equation and a new list of 30 points. The known lower bound has therefore increased from 29 to 30.

The contributors named Claude as one of those who found the curve. The registry provides the equation and the list of points needed to verify the result.