curve #2589

y² + xy = x³ − 223075912911370x + 1266758392652085092900

a-invariants: [1, 0, 0, -223075912911370, 1266758392652085092900]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 75752014470
discriminant (Δ): 17236939916172781832918641552507185216000000
Faltings height: 7.1013
naive height: 110.7292
primes of bad reduction: 2, 3, 5, 11, 19, 29, 31, 89, 151
regulator: 30.8141312198930208323799375499320982991320
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:11:15 UTC
last updated: 2026-09-26 20:11:15 UTC

Witness: 2 independent points log in to add more points →

(9418976, 1107074822)
(9550120, 2710622950)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-29, q=60; A=(3q-p)(p+q)^3=6226319, B=-(p+3q)(p-q)^3=106450319. 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 20:11:16 UTC · history