curve #2824

y² + y = x³ + x² − 23777x + 1402920

a-invariants: [0, 1, 1, -23777, 1402920]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/3ℤ
conductor (N): 413231
discriminant (Δ): 448479191069
Faltings height: 1.2135
naive height: 41.8431
primes of bad reduction: 7, 13, 19, 239
regulator: 19.343358243019390928932869330677215701986880
submitted by: David Renshaw
submitted at: 2026-09-26 20:24:04 UTC
last updated: 2026-09-26 20:24:04 UTC

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

(90, 9)
(166, 1434)
(-176, 408)

Commentary

Cremona 413231b1 = LMFDB 413231.b2: https://www.lmfdb.org/EllipticCurve/Q/413231/b/2 Mordell–Weil rank exactly 3 (Cremona's tables). Conductor 413231, minimal discriminant 448479191069, torsion ℤ/3ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:24:04 UTC · history