curve #116

y² + xy = x³ − 9227x + 340354

a-invariants: [1, 0, 0, -9227, 340354]
rank (lower bound): ≥ 6
conductor (N): 6822208199
naive height: 39.0033
Faltings height: 0.9339
discriminant (Δ): 6822208199 ★ record for rank ≥ 6
primes of bad reduction: 6822208199
regulator: 207.82607512733307132827843581134934911794454010300
submitted by: David Renshaw
last updated: 2026-07-01 22:59:17

Witness: 6 independent points

(55, -27)
(54, -10)
(57, -8)
(49, 56)
(62, 56)
(89, 431)

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 00:08:19 · history