Elliptic Curve Rank

The elliptic curve rank, also called the Mordell-Weil rank, of an elliptic curve over is the group rank of its group of rational points. By the Mordell-Weil theorem,

where is the finite subgroup of torsion points and the nonnegative integer is the rank (Silverman 1986). Equivalently, is the maximal number of linearly independent rational points of infinite order.

It is not known whether the ranks of elliptic curves over are bounded. A traditional conjecture says that they are unbounded. On the other hand, the heuristic of Park et al. (2019) predicts that they are bounded and that only finitely many elliptic curves over have rank greater than 21.

As of Aug. 2026, the largest known lower bound for the rank of an elliptic curve over is 30. The record elliptic curve, submitted to the ICARM Elliptic Curve Rank Leaderboard under the name "ranksunbounded" and subsequently attributed there to Claude working with L. Alpöge and A. Howell, is

Thirty linearly independent rational points have been exhibited, proving unconditionally that its rank is at least 30. The preceding records were rank at least 28, found by N. D. Elkies in 2006, and rank at least 29, found by Elkies and Z. Klagsbrun in 2024 (Dujella).

Applying the analytic method of Bober (2013) gives an upper bound of 31 under the generalized Riemann hypothesis for the multiplicity of as a zero of the elliptic curve's Hasse-Weil L-function. The sign in its functional equation is 1, so this multiplicity is an even number. Consequently, assuming both the generalized Riemann hypothesis and the Swinnerton-Dyer conjecture, the rank of this elliptic curve is exactly 30. Without these assumptions, only the lower bound of 30 is known. The largest rank of an elliptic curve whose exact rank is known unconditionally is 20 (Elkies and Klagsbrun 2020).