curve #2708

y² = x³ − 646571x + 200065370

a-invariants: [0, 0, 0, -646571, 200065370]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/4ℤ
conductor (N): 449072
discriminant (Δ): 8006561105813504
Faltings height: 2.0353
naive height: 51.7519
primes of bad reduction: 2, 13, 17, 127
regulator: 22.452312071989849225401070877668572098967919
submitted by: David Renshaw
submitted at: 2026-09-26 20:18:42 UTC
last updated: 2026-09-26 20:18:42 UTC

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

(445, 680)
(479, 510)
(-371, 19720)

Commentary

Cremona 449072l3 = LMFDB 449072.l1: https://www.lmfdb.org/EllipticCurve/Q/449072/l/1 Mordell–Weil rank exactly 3 (Cremona's tables). Conductor 449072, minimal discriminant 8006561105813504, torsion ℤ/4ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:18:42 UTC · history