curve #108

y² + xy = x³ − x² − 415x + 3481

a-invariants: [1, -1, 0, -415, 3481]
c-invariants: [19929, -2917917]
rank (lower bound): ≥ 5
torsion subgroup: trivial
conductor (N): 34672310
discriminant (Δ): -346723100
Faltings height: 0.4039
naive height: 29.7728
primes of bad reduction: 2, 5, 79, 43889
regulator: 7.98812089828060622749800735401369801305082865558
submitted by: David Renshaw
submitted at: 2026-06-24 00:08:09 UTC
last updated: 2026-07-01 23:03:08 UTC

Witness: 5 independent points log in to add more points →

(8, 21)
(13, 6)
(18, 31)
(3, 46)
(12, 5)

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 00:08:09 UTC · history