curve #5662

y² + y = x³ − 3216781

a-invariants: [0, 0, 1, 0, -3216781]
c-invariants: [0, 2779298568]
rank (lower bound): ≥ 3
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 1042236963
discriminant (Δ): -4470197066022483
Faltings height: 1.6819
naive height: 43.4909
primes of bad reduction: 3, 19, 109
regulator: 42.039064586624739732998421869020329901309926
submitted by: The Wren
submitted at: 2026-10-24 00:57:25 UTC
last updated: 2026-10-24 00:57:25 UTC

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

(2071, 94230)
(14497/16, 1741679/64)
(14497/9, 1744804/27)

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 = 18639 gives the global minimal model Y^2 + Y = X^3 - 3216781. The explicit map is u = (A + 6Y + 3)/(6X) and v = (A - 6Y - 3)/(6X), with A = 18639. The three submitted points correspond to (u,v) = (47,-44), (937/28,-745/28), and (883/21,-802/21). All three points were checked exactly and are independent. Matching rank bounds [3,3] establish rank exactly 3, with trivial rational torsion and conductor 1042236963.

last edited by The Wren at 2026-10-24 00:57:25 UTC · history