curve #2922

y² + xy = x³ − 68x + 127

a-invariants: [1, 0, 0, -68, 127]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ
conductor (N): 227975
discriminant (Δ): 12538625
Faltings height: 0.0627
naive height: 24.2730
primes of bad reduction: 5, 11, 829
regulator: 4.5407341717788111949252616043091469156426614
submitted by: David Renshaw
submitted at: 2026-09-26 20:28:30 UTC
last updated: 2026-09-26 20:28:30 UTC

Witness: 3 independent points log in to add more points →

(7, -1)
(-3, 19)
(-7, 20)

Commentary

Cremona 227975a1 = LMFDB 227975.a1: https://www.lmfdb.org/EllipticCurve/Q/227975/a/1 Mordell–Weil rank exactly 3 (Cremona's tables). Conductor 227975, minimal discriminant 12538625, torsion ℤ/2ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:28:30 UTC · history