curve #1170

y² + xy + y = x³ + x² − 152269x + 22806506

a-invariants: [1, 1, 1, -152269, 22806506]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 35139
discriminant (Δ): 208672635249 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 2
Faltings height: 1.5017
naive height: 47.4138
primes of bad reduction: 3, 13, 17, 53
regulator: 10.4683788541542612415751397791988801124289
submitted by: David Renshaw
submitted at: 2026-09-23 21:38:14 UTC
last updated: 2026-09-23 21:38:14 UTC

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

(56, 3774)
(-400, 4662)

Commentary

Cremona 35139c2, rank exactly 2 according to Cremona's tables. Witness generators from https://github.com/JohnCremona/ecdata/blob/master/allgens/allgens.30000-39999 . 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 21:38:14 UTC · history