curve #1110

y² + xy + y = x³ − x² + 4x + 6

a-invariants: [1, -1, 1, 4, 6]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/4ℤ
conductor (N): 117 ☆ record for ℤ/4ℤ torsion, rank ≥ 1
discriminant (Δ): -28431
Faltings height: -0.4622
naive height: 17.5108 ☆ record for ℤ/4ℤ torsion, rank ≥ 1
primes of bad reduction: 3, 13
regulator: 1.130335626179551006648752509880637923980
submitted by: David Renshaw
submitted at: 2026-09-23 20:24:17 UTC
last updated: 2026-09-23 20:24:17 UTC

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

(-1/4, 15/8)

Commentary

Cremona 117a1 (LMFDB 117.a): the smallest-conductor elliptic curve over Q with rank 1 and torsion subgroup Z/4Z, per Cremona's tables (PARI elldata search over conductors up to 600). Generator (-1/4, 15/8); the point (1, 2) has order 4.

last edited by David Renshaw at 2026-09-23 20:24:17 UTC · history