curve #1180

y² + xy = x³ − 1785x + 194922

a-invariants: [1, 0, 0, -1785, 194922]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/8ℤ
conductor (N): 1785
discriminant (Δ): -16074715228425
Faltings height: 1.2146
naive height: 37.8854 ☆ record for ℤ/8ℤ torsion, rank ≥ 1
primes of bad reduction: 3, 5, 7, 17
regulator: 2.023651471250951911811702097942194471646
submitted by: David Renshaw
submitted at: 2026-09-23 21:38:33 UTC
last updated: 2026-09-23 21:38:33 UTC

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

(63, 546)

Commentary

Cremona 1785o4, rank exactly 1 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 21:38:33 UTC · history