curve #3282

y² + xy + y = x³ − x² − 199x + 510

a-invariants: [1, -1, 1, -199, 510]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/4ℤ
conductor (N): 289
discriminant (Δ): 410338673
Faltings height: 0.3468
naive height: 27.4888
primes of bad reduction: 17
regulator: 3.035256177730332160832053603271747336052
submitted by: David Renshaw
submitted at: 2026-09-26 20:51:06 UTC
last updated: 2026-09-26 20:51:06 UTC

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

(-12, 38)

Commentary

Cremona 289a1 = LMFDB 289.a4: https://www.lmfdb.org/EllipticCurve/Q/289/a/4 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 289, minimal discriminant 410338673, torsion ℤ/4ℤ. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:51:06 UTC · history