curve #3003

y² + xy + y = x³ − x² + 149x + 3066

a-invariants: [1, -1, 1, 149, 3066]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/4ℤ
conductor (N): 51359
discriminant (Δ): -4374194671
Faltings height: 0.5308
naive height: 29.6038
primes of bad reduction: 7, 11, 23, 29
regulator: 10.0445113880872572176935395003776252499751
submitted by: David Renshaw
submitted at: 2026-09-26 20:32:51 UTC
last updated: 2026-09-26 20:32:51 UTC

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

(-2, 53)
(1116, 36713)

Commentary

Cremona 51359g1 = LMFDB 51359.e4: https://www.lmfdb.org/EllipticCurve/Q/51359/e/4 Mordell–Weil rank exactly 2 (Cremona's tables). Conductor 51359, minimal discriminant -4374194671, torsion ℤ/4ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:32:51 UTC · history