curve #3262

y² + y = x³ − 270030x + 54009000

a-invariants: [0, 0, 1, -270030, 54009000]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/3ℤ
conductor (N): 24003
discriminant (Δ): -216027
Faltings height: 1.3720
naive height: 49.1325
primes of bad reduction: 3, 7, 127
regulator: 47.8677123117664058247782421820818297523506
submitted by: David Renshaw
submitted at: 2026-09-26 20:47:53 UTC
last updated: 2026-09-26 20:47:53 UTC

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

(1257/4, 3439/8)
(-10986/25, 1171287/125)

Commentary

Cremona 24003f3 = LMFDB 24003.b1: https://www.lmfdb.org/EllipticCurve/Q/24003/b/1 Mordell–Weil rank exactly 2 (Cremona's tables). Conductor 24003, minimal discriminant -216027, torsion ℤ/3ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:47:53 UTC · history