curve #3228

y² = x³ + x² − 86x + 280

a-invariants: [0, 1, 0, -86, 280]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ
conductor (N): 2080
discriminant (Δ): 4160
Faltings height: -0.2429
naive height: 24.9883
primes of bad reduction: 2, 5, 13
regulator: 1.19405006443153500984712892878588038131964
submitted by: David Renshaw
submitted at: 2026-09-26 20:43:36 UTC
last updated: 2026-09-26 20:43:36 UTC

Witness: 2 independent points log in to add more points →

(6, 4)
(1, 14)

Commentary

Cremona 2080d1 = LMFDB 2080.a1: https://www.lmfdb.org/EllipticCurve/Q/2080/a/1 Mordell–Weil rank exactly 2 (Cremona's tables). Conductor 2080, minimal discriminant 4160, torsion ℤ/2ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:43:36 UTC · history