curve #3238

y² = x³ + x² + 96x + 196

a-invariants: [0, 1, 0, 96, 196]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ
conductor (N): 81328
discriminant (Δ): -67664896
Faltings height: 0.1910
naive height: 25.2962
primes of bad reduction: 2, 13, 17, 23
regulator: 3.5440189536549500648617999498029826487372104
submitted by: David Renshaw
submitted at: 2026-09-26 20:44:05 UTC
last updated: 2026-09-26 20:44:05 UTC

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

(2, 20)
(6, 32)
(-1, 10)

Commentary

Cremona 81328c1 = LMFDB 81328.b2: https://www.lmfdb.org/EllipticCurve/Q/81328/b/2 Mordell–Weil rank exactly 3 (Cremona's tables). Conductor 81328, minimal discriminant -67664896, torsion ℤ/2ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:44:05 UTC · history