curve #3325

y² + xy + y = x³ − 2166663x + 1227357656

a-invariants: [1, 0, 1, -2166663, 1227357656]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/6ℤ
conductor (N): 25870
discriminant (Δ): 16087906250
Faltings height: 1.9651
naive height: 55.3797
primes of bad reduction: 2, 5, 13, 199
regulator: 19.6282103989080732238258226901805611295247
submitted by: David Renshaw
submitted at: 2026-09-26 20:58:04 UTC
last updated: 2026-09-26 20:58:04 UTC

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

(1450, 32962)
(-1588, 26564)

Commentary

Cremona 25870b2 = LMFDB 25870.a2: https://www.lmfdb.org/EllipticCurve/Q/25870/a/2 Mordell–Weil rank exactly 2 (Cremona's tables). Conductor 25870, minimal discriminant 16087906250, torsion ℤ/6ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:58:04 UTC · history