curve #3106

y² + xy = x³ − x² − 198x + 1120

a-invariants: [1, -1, 0, -198, 1120]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 198
discriminant (Δ): 3175524
Faltings height: 0.1078
naive height: 27.4812
primes of bad reduction: 2, 3, 11
regulator: 0.3900859827946226837772726842820017242934
submitted by: David Renshaw
submitted at: 2026-09-26 20:37:33 UTC
last updated: 2026-09-26 20:37:33 UTC

Witness: 1 independent point log in to add more points →

(6, 8)

Commentary

Cremona 198a2 = LMFDB 198.a2: https://www.lmfdb.org/EllipticCurve/Q/198/a/2 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 198, minimal discriminant 3175524, torsion ℤ/2ℤ × ℤ/2ℤ. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:37:33 UTC · history