curve #3257

y² = x³ + x² − 258x + 1513

a-invariants: [0, 1, 0, -258, 1513]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/3ℤ
conductor (N): 12700
discriminant (Δ): -1270000
Faltings height: 0.1101
naive height: 28.2775
primes of bad reduction: 2, 5, 127
regulator: 1.65505017914477316253509993673317524096517
submitted by: David Renshaw
submitted at: 2026-09-26 20:47:02 UTC
last updated: 2026-09-26 20:47:02 UTC

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

(9, 1)
(12, 17)

Commentary

Cremona 12700i1 = LMFDB 12700.b1: https://www.lmfdb.org/EllipticCurve/Q/12700/b/1 Mordell–Weil rank exactly 2 (Cremona's tables). Conductor 12700, minimal discriminant -1270000, torsion ℤ/3ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:47:02 UTC · history