curve #75

y² + y = x³ + x² − 23846x + 1022562

a-invariants: [0, 1, 1, -23846, 1022562]
rank (lower bound): ≥ 8
conductor (N): 409086620841461
naive height: 41.8518
Faltings height: 1.5115 ★ record for rank ≥ 8
discriminant (Δ): 409086620841461 ★ record for rank ≥ 8
primes of bad reduction: 409086620841461
regulator: 13497.3934756072489297583974277373235085998065403862966
submitted by: David Renshaw
last updated: 2026-07-01 22:55:23

Witness: 8 independent points

(-5, 1068)
(-7, 1090)
(-15, 1173)
(22, 713)
(28, 614)
(37, 438)
(40, 366)
(47, 88)

Commentary

Provenance: Elkies-Watkins, "Elliptic curves of large rank and small conductor" (ANTS VI / arXiv:math/0403374), Table 4 low absolute-discriminant list for r=8: [0,1,1,-23846,1022562], |Delta|=409086620841461; their I-column records 78 integral x-coordinates. The submitted integral points certify rank >= 8 here.

last edited by David Renshaw at 2026-06-01 15:47:38 · history