curve #128

y² + xy = x³ − x² − 22221159x + 40791791609

a-invariants: [1, -1, 0, -22221159, 40791791609]
rank (lower bound): ≥ 7
conductor (N): 13077343449126
naive height: 62.3864
Faltings height: 3.0699
discriminant (Δ): -16408200601321750438668
primes of bad reduction: 2, 3, 11, 23, 419, 6853501
regulator: 244.7081168123110517424651740118236827089076400521766
submitted by: David Renshaw
last updated: 2026-07-01 22:57:21

Witness: 7 independent points

(3055, 36043)
(2296, 42115)
(778, 154447)
(1537, 100558)
(272, 186325)
(-3017, 284995)
(1240, 122437)

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 00:08:34 · history