curve #2312

y² + xy = x³ − x² − 858631155039x + 273407893746124473

a-invariants: [1, -1, 0, -858631155039, 273407893746124473]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 112342230
discriminant (Δ): 8220752981194140677239024315302562500
Faltings height: 5.8148
naive height: 94.0494
primes of bad reduction: 2, 3, 5, 7, 11, 13, 29, 43
regulator: 3.648235968938818284788972966801925519655
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:53:45 UTC
last updated: 2026-09-26 19:53:45 UTC

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

(-4194537/4, 1154733927/8)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-54, q=11; A=(3q-p)(p+q)^3=-6917109, B=-(p+3q)(p-q)^3=-5767125. 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 19:53:45 UTC · history