curve #5322

y² = x³ − 417083043

a-invariants: [0, 0, 0, 0, -417083043]
c-invariants: [0, 360359749152]
rank (lower bound): ≥ 4
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 135134905932
discriminant (Δ): -75149970375516414768
Faltings height: 2.4927
naive height: 53.2207
primes of bad reduction: 2, 3, 13, 907
regulator: 594.7376671857399797669783317517039691238051987
submitted by: David Renshaw
submitted at: 2026-10-21 13:09:09 UTC
last updated: 2026-10-21 13:09:09 UTC

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

(907, 18140)
(11791/9, 1155518/27)
(30433/16, 5145647/64)
(2275663/2601, 2108490202/132651)

Commentary

Cube-sum curve x^3 + y^3 = 212238 (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 13:09:09 UTC · history