curve #3269

y² = x³ + x² − 80344x − 7851484

a-invariants: [0, 1, 0, -80344, -7851484]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 85008
discriminant (Δ): 6743321251881984
Faltings height: 1.7687
naive height: 45.4958
primes of bad reduction: 2, 3, 7, 11, 23
regulator: 5.21580662531964353581616925654681600406768
submitted by: David Renshaw
submitted at: 2026-09-26 20:49:00 UTC
last updated: 2026-09-26 20:49:00 UTC

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

(344, 2310)
(470, 7644)

Commentary

Cremona 85008t3 = LMFDB 85008.bt3: https://www.lmfdb.org/EllipticCurve/Q/85008/bt/3 Mordell–Weil rank exactly 2 (Cremona's tables). Conductor 85008, minimal discriminant 6743321251881984, torsion ℤ/2ℤ × ℤ/4ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:49:00 UTC · history