curve #3217

y² + xy + y = x³ − x² − 11x + 2

a-invariants: [1, -1, 1, -11, 2]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ
conductor (N): 3717
discriminant (Δ): 78057
Faltings height: -0.3717
naive height: 18.7208
primes of bad reduction: 3, 7, 59
regulator: 0.925178410832445646886214715253508702948947
submitted by: David Renshaw
submitted at: 2026-09-26 20:43:08 UTC
last updated: 2026-09-26 20:43:08 UTC

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

(0, 1)
(-2, 4)

Commentary

Cremona 3717b1 = LMFDB 3717.a2: https://www.lmfdb.org/EllipticCurve/Q/3717/a/2 Mordell–Weil rank exactly 2 (Cremona's tables). Conductor 3717, minimal discriminant 78057, torsion ℤ/2ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:43:08 UTC · history