curve #3180

y² + xy + y = x³ − x² − 19230x − 943228

a-invariants: [1, -1, 1, -19230, -943228]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 326975
discriminant (Δ): 66820406640625
Faltings height: 1.3942
naive height: 41.2062
primes of bad reduction: 5, 11, 29, 41
regulator: 16.497949560267671766261881743318011837858891
submitted by: David Renshaw
submitted at: 2026-09-26 20:41:08 UTC
last updated: 2026-09-26 20:41:08 UTC

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

(-66, 220)
(-86, 305)
(184, 1220)

Commentary

Cremona 326975c2 = LMFDB 326975.c2: https://www.lmfdb.org/EllipticCurve/Q/326975/c/2 Mordell–Weil rank exactly 3 (Cremona's tables). Conductor 326975, minimal discriminant 66820406640625, torsion ℤ/2ℤ × ℤ/2ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:41:08 UTC · history