curve #3082

y² + y = x³ + x² + 9x + 9

a-invariants: [0, 1, 1, 9, 9]
rank (lower bound): ≥ 3
torsion subgroup: trivial
conductor (N): 58939
discriminant (Δ): -58939
Faltings height: -0.3969
naive height: 18.0921
primes of bad reduction: 17, 3467
regulator: 3.1813721975881424995385102317190892337351693
submitted by: David Renshaw
submitted at: 2026-09-26 20:36:25 UTC
last updated: 2026-09-26 20:36:25 UTC

Witness: 3 independent points log in to add more points →

(-1, 0)
(1, 4)
(-3/4, 9/8)

Commentary

Cremona 58939a1 = LMFDB 58939.a1: https://www.lmfdb.org/EllipticCurve/Q/58939/a/1 Mordell–Weil rank exactly 3 (Cremona's tables). Conductor 58939, minimal discriminant -58939, torsion trivial. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:36:25 UTC · history