curve #2686

y² + xy + y = x³ − x² − 2x

a-invariants: [1, -1, 1, -2, 0]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ
conductor (N): 99
discriminant (Δ): 297
Faltings height: -0.8353
naive height: 13.1833
primes of bad reduction: 3, 11
regulator: 0.3025713845614949991761430348115623877720
submitted by: David Renshaw
submitted at: 2026-09-26 20:17:43 UTC
last updated: 2026-09-26 20:17:43 UTC

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

(0, 0)

Commentary

Cremona 99a1 = LMFDB 99.a2: https://www.lmfdb.org/EllipticCurve/Q/99/a/2 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 99, minimal discriminant 297, torsion ℤ/2ℤ. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:17:43 UTC · history