curve #3302

y² + xy + y = x³ + x² − 78802000x + 269216088242

a-invariants: [1, 1, 1, -78802000, 269216088242]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/4ℤ
conductor (N): 19065
discriminant (Δ): 14775375
Faltings height: 2.6419
naive height: 66.1609
primes of bad reduction: 3, 5, 31, 41
regulator: 28.8201159417567885467655772298523162140052
submitted by: David Renshaw
submitted at: 2026-09-26 20:54:20 UTC
last updated: 2026-09-26 20:54:20 UTC

Witness: 2 independent points log in to add more points →

(6117, 123886)
(28963/4, 2209853/8)

Commentary

Cremona 19065g4 = LMFDB 19065.c1: https://www.lmfdb.org/EllipticCurve/Q/19065/c/1 Mordell–Weil rank exactly 2 (Cremona's tables). Conductor 19065, minimal discriminant 14775375, torsion ℤ/4ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:54:20 UTC · history