curve #111

y² + y = x³ − 139x + 732

a-invariants: [0, 0, 1, -139, 732]
rank (lower bound): ≥ 5
conductor (N): 59754491
naive height: 26.7154
Faltings height: 0.2148
discriminant (Δ): -59754491
primes of bad reduction: 409, 146099
regulator: 21.6690124194189046816235098629475654280180199560
submitted by: David Renshaw
last updated: 2026-07-01 23:01:14

Witness: 5 independent points

(8, 11)
(4, 15)
(9, 14)
(3, 18)
(10, 18)

Commentary

From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).

last edited by David Renshaw at 2026-06-24 00:08:13 · history