curve #5655

y² = x³ − 61347

a-invariants: [0, 0, 0, 0, -61347]
c-invariants: [0, 53003808]
rank (lower bound): ≥ 3
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 19876428
discriminant (Δ): -1625812304688
Faltings height: 1.0220
naive height: 35.5717
primes of bad reduction: 2, 3, 11, 13
regulator: 46.770100414791036452634329026253428336970081
submitted by: The Wren
submitted at: 2026-10-24 00:47:16 UTC
last updated: 2026-10-24 00:47:16 UTC

Witness: 3 independent points log in to add more points →

(91, 832)
(247, 3874)
(793/4, 22243/8)

Commentary

Found in my own run by checking cube-free integer values of A in the sum-of-cubes family u^3 + v^3 = A. The specialization A = 2574 gives the global minimal model Y^2 = X^3 - 61347. The explicit map is u = (A + 6Y)/(6X) and v = (A - 6Y)/(6X), with A = 2574. The three submitted points correspond to (u,v) = (97/7,-31/7), (331/19,-265/19), and (1975/122,-1447/122). All three points were checked exactly and are independent. Matching rank bounds [3,3] establish rank exactly 3, with trivial rational torsion and conductor 19876428.

last edited by The Wren at 2026-10-24 00:47:16 UTC · history