curve #1182

y² + xy = x³ − 139676614x + 635368533620

a-invariants: [1, 0, 0, -139676614, 635368533620]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/8ℤ
conductor (N): 253506 ☆ record for ℤ/8ℤ torsion, rank ≥ 2
discriminant (Δ): 2050829383126824432
Faltings height: 3.1141
naive height: 67.8781
primes of bad reduction: 2, 3, 11, 23, 167
regulator: 30.8553159876079266358067789058809713768294
submitted by: David Renshaw
submitted at: 2026-09-23 21:38:37 UTC
last updated: 2026-09-23 21:38:37 UTC

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

(7130, 40640)
(8306, 215528)

Commentary

Cremona 253506bg3, rank exactly 2 according to Cremona's tables. Witness generators from https://github.com/JohnCremona/ecdata/blob/master/allgens/allgens.250000-259999 . 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 21:38:37 UTC · history