curve #2641

y² + xy + y = x³ − 57632074783x + 5044133965815806

a-invariants: [1, 0, 1, -57632074783, 5044133965815806]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 124061070
discriminant (Δ): 1259512682759418221306581370062500
Faltings height: 5.1043
naive height: 85.9456
primes of bad reduction: 2, 3, 5, 7, 17, 19, 31, 59
regulator: 10.9805377637367959501846906225339610080605
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:14:45 UTC
last updated: 2026-09-26 20:14:45 UTC

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

(-240203, 71031578)
(-65735, 92491142)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-38, q=7; A=(3q-p)(p+q)^3=-1757669, B=-(p+3q)(p-q)^3=-1549125. 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 20:14:45 UTC · history