curve #1129

y² + xy = x³ + x² − 42x

a-invariants: [1, 1, 0, -42, 0]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ
conductor (N): 113946
discriminant (Δ): 4785732
Faltings height: -0.0281 ☆ record for ℤ/2ℤ torsion, rank ≥ 3
naive height: 22.8636 ☆ record for ℤ/2ℤ torsion, rank ≥ 3
primes of bad reduction: 2, 3, 7, 2713
regulator: 1.9255937080775220743267207961480461492800741
submitted by: David Renshaw
submitted at: 2026-09-23 21:36:57 UTC
last updated: 2026-09-23 21:36:57 UTC

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

(6, 0)
(7, 7)
(-3, -9)

Commentary

Cremona 113946b1, rank exactly 3 according to Cremona's tables. Witness generators from https://github.com/JohnCremona/ecdata/blob/master/allgens/allgens.110000-119999 . 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 21:36:57 UTC · history