curve #99

y² + xy = x³ − x² − 10194109x + 12647638369

a-invariants: [1, -1, 0, -10194109, 12647638369]
rank (lower bound): ≥ 10
torsion subgroup: trivial
conductor (N): 22006161865320788846
naive height: 60.0442
Faltings height: 2.8655
discriminant (Δ): -1276357388188605753068
primes of bad reduction: 2, 29, 272446637, 1392627151
regulator: 303375.5907842359104425786052982343182424858155554328186436
submitted by: David Renshaw
submitted at: 2026-06-23 23:50:29 UTC
last updated: 2026-07-01 22:50:32 UTC

Witness: 10 independent points

(1740, 12383)
(1566, 22069)
(1392, 33263)
(1334, 37033)
(1943, 12180)
(1653, 16820)
(2784, 75023)
(2233, 30740)
(3190, 110577)
(1015, 57333)

Commentary

EW 2004 (Elkies-Watkins, ANTS-VI, arXiv:math/0403374) Table 2 low-conductor rank-10 list; this curve was absent from the leaderboard. Rank-10 witness via 10 independent rational points (Neron-Tate pairing).

last edited by David Renshaw at 2026-06-23 23:50:29 UTC · history