curve #2638

y² + xy + y = x³ − 86619196599x + 9572863706784166

a-invariants: [1, 0, 1, -86619196599, 9572863706784166]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 459900870
discriminant (Δ): 2004790404876543590819949689724900
Faltings height: 5.1693
naive height: 87.1680
primes of bad reduction: 2, 3, 5, 11, 13, 23, 59, 79
regulator: 12.2465390663756417614454192298079326712657
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:14:31 UTC
last updated: 2026-09-26 20:14:31 UTC

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

(-314959, 75062166)
(-1093813/4, 906588399/8)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-10, q=23; A=(3q-p)(p+q)^3=173563, B=-(p+3q)(p-q)^3=2120283. 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 20:14:31 UTC · history