curve #1138

y² + xy + y = x³ − x² − 41x + 96

a-invariants: [1, -1, 1, -41, 96]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 117 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 1
discriminant (Δ): 1108809
Faltings height: -0.1156
naive height: 22.7314
primes of bad reduction: 3, 13
regulator: 0.5651678130897755033243762549403189619899
submitted by: David Renshaw
submitted at: 2026-09-23 21:37:19 UTC
last updated: 2026-09-23 21:37:19 UTC

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

(6, 3)

Commentary

Cremona 117a2, 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 21:37:19 UTC · history