curve #2204

y² + xy = x³ − 69172920966290x + 134553503911665092100

a-invariants: [1, 0, 0, -69172920966290, 134553503911665092100]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 43127978790
discriminant (Δ): 13361834109079990454820368569429852224000000
Faltings height: 6.9755
naive height: 107.2165
primes of bad reduction: 2, 3, 5, 13, 47, 73, 167, 193
regulator: 5.832860558377862359214034445383864412914
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:46:55 UTC
last updated: 2026-09-26 19:46:55 UTC

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

(7375780, 5056689310)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-13, q=60; A=(3q-p)(p+q)^3=20037839, B=-(p+3q)(p-q)^3=64965839. 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 19:46:55 UTC · history