curve #135

y² + xy + y = x³ − 1076185x + 496031340

a-invariants: [1, 0, 1, -1076185, 496031340]
rank (lower bound): ≥ 9
conductor (N): 77211251506212554
naive height: 53.5678
Faltings height: 2.4508
discriminant (Δ): -26560670518137118576
primes of bad reduction: 2, 43, 897805250072239
regulator: 53538.096568535612150394273481249423219180410009474496562
submitted by: David Renshaw
last updated: 2026-07-01 22:52:28

Witness: 9 independent points

(679, 8518)
(335, 12990)
(1023, 21074)
(-525, 30534)
(-1041, 22622)
(-1213, 4734)
(120, 19139)
(34, 21418)
(1668, 56979)

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 00:08:43 · history