curve #2214

y² + xy + y = x³ − 86158437134683x + 307724834828804204306

a-invariants: [1, 0, 1, -86158437134683, 307724834828804204306]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 6060428430
discriminant (Δ): 24926681213789409854798694891246162562500
Faltings height: 6.7214
naive height: 107.8752
primes of bad reduction: 2, 3, 5, 7, 17, 83, 113, 181
regulator: 6.109352244790208259350345762478860590032
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:47:33 UTC
last updated: 2026-09-26 19:47:33 UTC

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

(5436790, 59559617)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-34, q=49; A=(3q-p)(p+q)^3=610875, B=-(p+3q)(p-q)^3=64611931. 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 19:47:33 UTC · history