curve #2310

y² + xy = x³ − 1085603747670x + 419658188452164900

a-invariants: [1, 0, 0, -1085603747670, 419658188452164900]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 67038270
discriminant (Δ): 5802196644186120174962586157056000000
Faltings height: 5.8198
naive height: 94.7531
primes of bad reduction: 2, 3, 5, 13, 19, 83, 109
regulator: 4.507700772265674674578462421650187148828
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:53:37 UTC
last updated: 2026-09-26 19:53:37 UTC

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

(-1156140, 360303930)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-13, q=32; A=(3q-p)(p+q)^3=747631, B=-(p+3q)(p-q)^3=7563375. 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 19:53:38 UTC · history