curve #2603

y² + xy = x³ − 10117614064636x − 11952705719815609840

a-invariants: [1, 0, 0, -10117614064636, -11952705719815609840]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 1710215130
discriminant (Δ): 4566243186188487375650642503023096729600
Faltings height: 6.3765
naive height: 101.4495
primes of bad reduction: 2, 3, 5, 13, 17, 31, 53, 157
regulator: 52.7593226214429980003418049410827369080489
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:12:08 UTC
last updated: 2026-09-26 20:12:08 UTC

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

(-2092516, 238165556)
(3707918, 1227334016)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-1, q=52; A=(3q-p)(p+q)^3=20826207, B=-(p+3q)(p-q)^3=23075935. 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 20:12:09 UTC · history