curve #2219

y² + xy = x³ − 34188488874936x + 76745045668917072960

a-invariants: [1, 0, 0, -34188488874936, 76745045668917072960]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 4929154230
discriminant (Δ): 13129170590135535878354008725791805542400
Faltings height: 6.5676
naive height: 105.1023
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 53, 163
regulator: 3.417315333404163390631058587516207885877
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:47:51 UTC
last updated: 2026-09-26 19:47:51 UTC

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

(3222768, 187865736)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-55, q=36; A=(3q-p)(p+q)^3=-1118017, B=-(p+3q)(p-q)^3=39939263. The listed points are independent modulo torsion by the exact 2-descent Kummer squareclass map. The rank is a proven lower bound; no upper bound is claimed.

last edited by Natanael Wildner Fraga at 2026-09-26 19:47:51 UTC · history