curve #1131

y² = x³ − x² − 835x + 9571

a-invariants: [0, -1, 0, -835, 9571]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ
conductor (N): 80256
discriminant (Δ): 160512 ☆ record for ℤ/2ℤ torsion, rank ≥ 3
Faltings height: 0.2377
naive height: 31.7971
primes of bad reduction: 2, 3, 11, 19
regulator: 6.6551076105186147677124242039055541907973747
submitted by: David Renshaw
submitted at: 2026-09-23 21:37:01 UTC
last updated: 2026-09-23 21:37:01 UTC

Witness: 3 independent points log in to add more points →

(18, 7)
(9, 52)
(26, 69)

Commentary

Cremona 80256a1, rank exactly 3 according to Cremona's tables. Witness generators from https://github.com/JohnCremona/ecdata/blob/master/allgens/allgens.80000-89999 . 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 21:37:01 UTC · history