curve #2285

y² + xy = x³ − 2312057375586x + 980849018884859460

a-invariants: [1, 0, 0, -2312057375586, 980849018884859460]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 67458930
discriminant (Δ): 375386591414073955102271054843904000000
Faltings height: 6.1094
naive height: 97.0211
primes of bad reduction: 2, 3, 5, 7, 11, 19, 29, 53
regulator: 4.595283073232922991952574108884616002349
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:52:03 UTC
last updated: 2026-09-26 19:52:03 UTC

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

(1233036, 67793670)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-53, q=24; A=(3q-p)(p+q)^3=-3048625, B=-(p+3q)(p-q)^3=8674127. 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 19:52:03 UTC · history