curve #1021

y² + xy = x³ − 429x + 3384

a-invariants: [1, 0, 0, -429, 3384]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 429
discriminant (Δ): 1656369 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 1
Faltings height: 0.1945 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 1
naive height: 29.7981
primes of bad reduction: 3, 11, 13
regulator: 2.115453196439503695102406459953893088052
submitted by: David Renshaw
submitted at: 2026-09-23 15:16:30 UTC
last updated: 2026-09-23 15:16:30 UTC

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

(24, 72)

Commentary

Cremona 429b2 = LMFDB 429.b4: https://www.lmfdb.org/EllipticCurve/Q/429/b/4 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 429, minimal discriminant 1656369, torsion ℤ/2ℤ × ℤ/4ℤ. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-23 15:16:30 UTC · history