curve #3224

y² + xy = x³ + x² − 8x + 12

a-invariants: [1, 1, 0, -8, 12]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ
conductor (N): 2130
discriminant (Δ): -63900
Faltings height: -0.3814
naive height: 19.0020
primes of bad reduction: 2, 3, 5, 71
regulator: 0.342631423899141355230254981426106780108122
submitted by: David Renshaw
submitted at: 2026-09-26 20:43:26 UTC
last updated: 2026-09-26 20:43:26 UTC

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

(1, 2)
(6, 12)

Commentary

Cremona 2130b1 = LMFDB 2130.a2: https://www.lmfdb.org/EllipticCurve/Q/2130/a/2 Mordell–Weil rank exactly 2 (Cremona's tables). Conductor 2130, minimal discriminant -63900, torsion ℤ/2ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:43:26 UTC · history