curve #646

y² + xy = x³ − x

a-invariants: [1, 0, 0, -1, 0]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ
conductor (N): 65 ☆ record for ℤ/2ℤ torsion, rank ≥ 1
discriminant (Δ): 65 ☆ record for ℤ/2ℤ torsion, rank ≥ 1
Faltings height: -0.9616 ☆ record for ℤ/2ℤ torsion, rank ≥ 1
naive height: 11.6755 ☆ record for ℤ/2ℤ torsion, rank ≥ 1
primes of bad reduction: 5, 13
regulator: 0.3755140986612663218044728768245783176621
submitted by: David Renshaw
submitted at: 2026-09-08 19:03:16 UTC
last updated: 2026-09-08 19:03:16 UTC

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

(1, 0)

Commentary

Cremona 65a1 = LMFDB 65.a1. Curve data, generator (1, 0) and rank from the LMFDB: https://www.lmfdb.org/EllipticCurve/Q/65/a/1 This curve and the 2-isogenous 65a2 are the curves of smallest conductor that have both positive rank and nontrivial torsion.

last edited by NDElkies at 2026-09-09 02:53:49 UTC · history