curve #2309

y² + xy + y = x³ − 1087135728103x + 432611371911566006

a-invariants: [1, 0, 1, -1087135728103, 432611371911566006]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 2652638070
discriminant (Δ): 1380230807215086096924461522468062500
Faltings height: 5.7542
naive height: 94.7573
primes of bad reduction: 2, 3, 5, 19, 23, 31, 61, 107
regulator: 5.169213186871413313596942653641559016341
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:53:34 UTC
last updated: 2026-09-26 19:53:34 UTC

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

(-1132130, 461346797)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-38, q=23; A=(3q-p)(p+q)^3=-361125, B=-(p+3q)(p-q)^3=7036411. 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 19:53:34 UTC · history