curve #3243

y² = x³ + x² − 5123x + 139441

a-invariants: [0, 1, 0, -5123, 139441]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ
conductor (N): 491776
discriminant (Δ): 983552
Faltings height: 0.6070
naive height: 37.2383
primes of bad reduction: 2, 17, 113
regulator: 14.966039429849997703524055888971327115296011
submitted by: David Renshaw
submitted at: 2026-09-26 20:44:44 UTC
last updated: 2026-09-26 20:44:44 UTC

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

(39, 22)
(49, 92)
(319/9, 1630/27)

Commentary

Cremona 491776a1 = LMFDB 491776.a2: https://www.lmfdb.org/EllipticCurve/Q/491776/a/2 Mordell–Weil rank exactly 3 (Cremona's tables). Conductor 491776, minimal discriminant 983552, torsion ℤ/2ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:44:44 UTC · history