curve #137

y² + xy = x³ − x² − 41032399x + 106082399089

a-invariants: [1, -1, 0, -41032399, 106082399089]
c-invariants: [1969555161, -91646329814685]
rank (lower bound): ≥ 11
torsion subgroup: trivial
conductor (N): 219576020293485812169274
discriminant (Δ): -439152040586971624338548
Faltings height: 3.2932
naive height: 64.2979
primes of bad reduction: 2, 9913609, 11074474507391093
regulator: 9786539.43783265571672432007979536225451355144124014541976368
submitted by: David Renshaw
submitted at: 2026-06-24 00:08:45 UTC
last updated: 2026-07-01 22:48:34 UTC

Witness: 11 independent points log in to add more points →

(3790, 68807)
(3342, 77515)
(3996, 74899)
(4238, 88935)
(2940, 102707)
(2878, 107295)
(4382, 99823)
(2764, 116003)
(2756, 116625)
(2672, 123225)
(2660, 124177)

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 00:08:45 UTC · history