curve #5323

y² = x³ − 500236707

a-invariants: [0, 0, 0, 0, -500236707]
c-invariants: [0, 432204514848]
rank (lower bound): ≥ 4
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 162076693068
discriminant (Δ): -108102281629048062768
Faltings height: 2.5230
naive height: 53.5843
primes of bad reduction: 2, 3, 37, 349
regulator: 619.7925933893859668674832839842741446136363122
submitted by: David Renshaw
submitted at: 2026-10-21 13:09:14 UTC
last updated: 2026-10-21 13:09:14 UTC

Witness: 4 independent points log in to add more points →

(2443, 118660)
(5143, 368150)
(12913/16, 322825/64)
(26899/25, 3412732/125)

Commentary

Cube-sum curve x^3 + y^3 = 232434 (Weierstrass model y^2 = x^3 - 432k^2, minimal model shown). Found in an exhaustive scan of cube-free k <= 7*10^5 by minimal-model naive height, with a Mestre-Nagao a_p sieve (p = 1 mod 3) and PARI ellrank 2-descent; companion to #5313 (k = 111690).

last edited by David Renshaw at 2026-10-21 13:09:14 UTC · history