curve #49

y² + y = x³ − 16312387x + 25970162646

a-invariants: [0, 0, 1, -16312387, 25970162646]
rank (lower bound): ≥ 10
torsion subgroup: trivial
conductor (N): 10189285026863130793 ★ record for rank ≥ 10
naive height: 61.4836
Faltings height: 3.0269
discriminant (Δ): -13561938370754827085483
primes of bad reduction: 11, 1297, 4423, 75707, 2132839
regulator: 990834.3392128087320646545893821685852413506370864254350345
submitted by: David Renshaw
submitted at: 2026-05-28 18:40:50 UTC
last updated: 2026-07-01 22:50:32 UTC

Witness: 10 independent points

(-4674, 10235)
(-4036, 161441)
(-2413, 226455)
(-305, 175832)
(1277, 84980)
(-4509, 88599)
(2463, 27096)
(3246, 84980)
(4687, 229080)
(7993, 637371)

Commentary

Smallest known conductor of an elliptic curve with rank ≥ 10 (Elkies–Watkins, 2004). From their paper "Elliptic Curves of Large Rank and Small Conductor" (arXiv:math/0403374).

last edited by David Renshaw at 2026-05-28 18:41:35 UTC · history