curve #3276

y² + xy = x³ − 24x + 63

a-invariants: [1, 0, 0, -24, 63]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/4ℤ
conductor (N): 429
discriminant (Δ): -938223
Faltings height: -0.1521
naive height: 21.8720
primes of bad reduction: 3, 11, 13
regulator: 1.057726598219751847551203229976946544026
submitted by: David Renshaw
submitted at: 2026-09-26 20:50:08 UTC
last updated: 2026-09-26 20:50:08 UTC

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

(6, 9)

Commentary

Cremona 429b1 = LMFDB 429.b5: https://www.lmfdb.org/EllipticCurve/Q/429/b/5 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 429, minimal discriminant -938223, torsion ℤ/4ℤ. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:50:09 UTC · history