curve #3234

y² + xy = x³ − 306x + 2035

a-invariants: [1, 0, 0, -306, 2035]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ
conductor (N): 14705
discriminant (Δ): 14705
Faltings height: 0.0026
naive height: 28.7846
primes of bad reduction: 5, 17, 173
regulator: 3.38731966177852374718933070726322964981873
submitted by: David Renshaw
submitted at: 2026-09-26 20:43:56 UTC
last updated: 2026-09-26 20:43:56 UTC

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

(9, 1)
(31/4, 65/8)

Commentary

Cremona 14705a1 = LMFDB 14705.a1: https://www.lmfdb.org/EllipticCurve/Q/14705/a/1 Mordell–Weil rank exactly 2 (Cremona's tables). Conductor 14705, minimal discriminant 14705, torsion ℤ/2ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:43:56 UTC · history