curve #3263

y² + xy = x³ − x² − 472197x + 125009643

a-invariants: [1, -1, 0, -472197, 125009643]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/3ℤ
conductor (N): 328482
discriminant (Δ): 328482
Faltings height: 1.4950
naive height: 50.8091
primes of bad reduction: 2, 3, 7, 11, 79
regulator: 93.2803951522359627041126609975247237397707
submitted by: David Renshaw
submitted at: 2026-09-26 20:48:02 UTC
last updated: 2026-09-26 20:48:02 UTC

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

(-2573/4, 104485/8)
(650623/1764, 57436063/74088)

Commentary

Cremona 328482o1 = LMFDB 328482.o3: https://www.lmfdb.org/EllipticCurve/Q/328482/o/3 Mordell–Weil rank exactly 2 (Cremona's tables). Conductor 328482, minimal discriminant 328482, torsion ℤ/3ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:48:02 UTC · history