curve #50

y² + y = x³ − 16359067x + 26274178986

a-invariants: [0, 0, 1, -16359067, 26274178986]
rank (lower bound): ≥ 11
torsion subgroup: trivial
conductor (N): 18031737725935636520843 ★ record for rank ≥ 11
naive height: 61.5069 ★ record for rank ≥ 11
Faltings height: 3.0412 ★ record for rank ≥ 11
discriminant (Δ): -18031737725935636520843 ★ record for rank ≥ 11
primes of bad reduction: 18031737725935636520843
regulator: 28087940.2288126473904406024985250980292392927967906350917450
submitted by: David Renshaw
submitted at: 2026-05-28 18:41:24 UTC
last updated: 2026-07-01 22:48:33 UTC

Witness: 11 independent points

(-4683, 13535)
(-4657, 38196)
(-3091, 217503)
(-826, 198048)
(1165, 93792)
(1921, 44015)
(2315, 28452)
(2837, 51935)
(3656, 123825)
(5753, 350096)
(9565, 863072)

Commentary

Smallest known conductor of an elliptic curve with rank ≥ 11 (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 18:41:36 UTC · history