curve #1139

y² + xy + y = x³ + x² − 71x − 196

a-invariants: [1, 1, 1, -71, -196]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 3315
discriminant (Δ): 10989225 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 2
Faltings height: 0.0584 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 2
naive height: 24.4025 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 2
primes of bad reduction: 3, 5, 13, 17
regulator: 1.35634345933378764540912298034212488754749
submitted by: David Renshaw
submitted at: 2026-09-23 21:37:20 UTC
last updated: 2026-09-23 21:37:20 UTC

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

(-4, 8)
(22, 86)

Commentary

Cremona 3315b2, 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 21:37:20 UTC · history