curve #1156

y² = x³ − x² − 39x + 264

a-invariants: [0, -1, 0, -39, 264]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/4ℤ
conductor (N): 5304
discriminant (Δ): -23305776
Faltings height: 0.0997 ☆ record for ℤ/4ℤ torsion, rank ≥ 2
naive height: 24.5735 ☆ record for ℤ/4ℤ torsion, rank ≥ 2
primes of bad reduction: 2, 3, 13, 17
regulator: 3.96846021424446964617745345128244670643630
submitted by: David Renshaw
submitted at: 2026-09-23 21:37:50 UTC
last updated: 2026-09-23 21:37:50 UTC

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

(1, 15)
(8, 20)

Commentary

Cremona 5304k1, rank exactly 2 according to Cremona's tables. Witness generators from https://github.com/JohnCremona/ecdata/blob/master/allgens/allgens.00000-09999 . Found by screening the local allgens tables for missing torsion size records; torsion and independence independently verified by the site.

last edited by David Renshaw at 2026-09-23 21:37:50 UTC · history