curve #3102

y² + xy = x³ − x² − 19x

a-invariants: [1, -1, 0, -19, 0]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 665
discriminant (Δ): 442225
Faltings height: -0.2267
naive height: 20.4764
primes of bad reduction: 5, 7, 19
regulator: 2.074054851350538613018854330110751035362
submitted by: David Renshaw
submitted at: 2026-09-26 20:37:24 UTC
last updated: 2026-09-26 20:37:24 UTC

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

(16, 52)

Commentary

Cremona 665b2 = LMFDB 665.e2: https://www.lmfdb.org/EllipticCurve/Q/665/e/2 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 665, minimal discriminant 442225, torsion ℤ/2ℤ × ℤ/2ℤ. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:37:24 UTC · history