curve #1113

y² + xy = x³ − 11619x + 462024

a-invariants: [1, 0, 0, -11619, 462024]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 14763 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 2
discriminant (Δ): 7785255102849
Faltings height: 1.2360
naive height: 39.6948
primes of bad reduction: 3, 7, 19, 37
regulator: 3.28026810609458732173296481990354897968927
submitted by: David Renshaw
submitted at: 2026-09-23 20:39:54 UTC
last updated: 2026-09-23 20:39:54 UTC

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

(-117, 531)
(-75, 993)

Commentary

Cremona 14763h2: the smallest-conductor elliptic curve over Q with rank 2 and torsion subgroup Z/2Z x Z/4Z, per Cremona's tables (PARI elldata scan of every curve with this torsion up to conductor 14763; ellrank proves rank exactly 2 and no smaller-conductor case was left inconclusive). Generators (-117, 531) and (-75, 993).

last edited by David Renshaw at 2026-09-23 20:39:54 UTC · history