curve #3270

y² = x³ − x² − 126404x − 13988796

a-invariants: [0, -1, 0, -126404, -13988796]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 50232
discriminant (Δ): 44213957363407104
Faltings height: 1.9104
naive height: 46.8553
primes of bad reduction: 2, 3, 7, 13, 23
regulator: 6.79046028776988483926565546408697567992636
submitted by: David Renshaw
submitted at: 2026-09-26 20:49:10 UTC
last updated: 2026-09-26 20:49:10 UTC

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

(-176, 1666)
(494, 6624)

Commentary

Cremona 50232r2 = LMFDB 50232.c5: https://www.lmfdb.org/EllipticCurve/Q/50232/c/5 Mordell–Weil rank exactly 2 (Cremona's tables). Conductor 50232, minimal discriminant 44213957363407104, torsion ℤ/2ℤ × ℤ/4ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:49:10 UTC · history