curve #3241

y² + xy + y = x³ + x² − 1935x + 31956

a-invariants: [1, 1, 1, -1935, 31956]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ
conductor (N): 278499
discriminant (Δ): 835497
Faltings height: 0.4262
naive height: 34.3172
primes of bad reduction: 3, 13, 37, 193
regulator: 7.3351883682610550005877424535280515111017028
submitted by: David Renshaw
submitted at: 2026-09-26 20:44:25 UTC
last updated: 2026-09-26 20:44:25 UTC

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

(24, -4)
(29, 21)
(13, 89)

Commentary

Cremona 278499a1 = LMFDB 278499.a2: https://www.lmfdb.org/EllipticCurve/Q/278499/a/2 Mordell–Weil rank exactly 3 (Cremona's tables). Conductor 278499, minimal discriminant 835497, torsion ℤ/2ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:44:25 UTC · history