curve #1130

y² + xy + y = x³ − 131x + 558

a-invariants: [1, 0, 1, -131, 558]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ
conductor (N): 59450 ☆ record for ℤ/2ℤ torsion, rank ≥ 3
discriminant (Δ): 2378000
Faltings height: 0.0433
naive height: 26.2282
primes of bad reduction: 2, 5, 29, 41
regulator: 2.5256934463377690704964308580045933536470083
submitted by: David Renshaw
submitted at: 2026-09-23 21:36:58 UTC
last updated: 2026-09-23 21:36:58 UTC

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

(3, 12)
(6, -4)
(8, 2)

Commentary

Cremona 59450i1, rank exactly 3 according to Cremona's tables. Witness generators from https://github.com/JohnCremona/ecdata/blob/master/allgens/allgens.50000-59999 . 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 21:36:58 UTC · history