curve #3137

y² = x³ − x² − 224x − 816

a-invariants: [0, -1, 0, -224, -816]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 9744
discriminant (Δ): 379782144
Faltings height: 0.3512
naive height: 27.8530
primes of bad reduction: 2, 3, 7, 29
regulator: 1.88643100546349253788780218449459888751273
submitted by: David Renshaw
submitted at: 2026-09-26 20:39:00 UTC
last updated: 2026-09-26 20:39:00 UTC

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

(24, 84)
(-8, 20)

Commentary

Cremona 9744c2 = LMFDB 9744.c2: https://www.lmfdb.org/EllipticCurve/Q/9744/c/2 Mordell–Weil rank exactly 2 (Cremona's tables). Conductor 9744, minimal discriminant 379782144, torsion ℤ/2ℤ × ℤ/2ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:39:00 UTC · history