curve #3284

y² + xy = x³ − 6864x + 218313

a-invariants: [1, 0, 0, -6864, 218313]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/4ℤ
conductor (N): 429
discriminant (Δ): 1287
Faltings height: 0.5411
naive height: 38.1157
primes of bad reduction: 3, 11, 13
regulator: 4.230906392879007390204812919907786176104
submitted by: David Renshaw
submitted at: 2026-09-26 20:51:26 UTC
last updated: 2026-09-26 20:51:26 UTC

Witness: 1 independent point log in to add more points →

(104, 743)

Commentary

Cremona 429b4 = LMFDB 429.b1: https://www.lmfdb.org/EllipticCurve/Q/429/b/1 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 429, minimal discriminant 1287, torsion ℤ/4ℤ. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:51:26 UTC · history