curve #2220

y² + xy + y = x³ − 31418952341393x + 67189157039531583056

a-invariants: [1, 0, 1, -31418952341393, 67189157039531583056]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 33265364370
discriminant (Δ): 34761610203654592112422136346610020562500
Faltings height: 6.5970
naive height: 104.8489
primes of bad reduction: 2, 3, 5, 7, 23, 29, 31, 47, 163
regulator: 9.394482581598922725045295142619066416311
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:47:55 UTC
last updated: 2026-09-26 19:47:55 UTC

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

(-6443590, 1453432407)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-58, q=35; A=(3q-p)(p+q)^3=-1983221, B=-(p+3q)(p-q)^3=37804779. 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 19:47:55 UTC · history