curve #2653

y² + xy + y = x³ − 61980084563x + 5938612083878906

a-invariants: [1, 0, 1, -61980084563, 5938612083878906]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 154392810
discriminant (Δ): 2878646086865277774574520062500
Faltings height: 4.8742
naive height: 86.1638
primes of bad reduction: 2, 3, 5, 11, 13, 17, 29, 73
regulator: 7.66969307670939264978926766790812977553034
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:15:41 UTC
last updated: 2026-09-26 20:15:41 UTC

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

(-899925/4, 738255371/8)
(-192660, 103675162)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-22, q=17; A=(3q-p)(p+q)^3=-9125, B=-(p+3q)(p-q)^3=1720251. 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 20:15:42 UTC · history