curve #5324

y² + y = x³ − 520070167

a-invariants: [0, 0, 1, 0, -520070167]
c-invariants: [0, 449340624072]
rank (lower bound): ≥ 4
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 168502734027
discriminant (Δ): -116844326644337052003
Faltings height: 2.5295
naive height: 53.6621
primes of bad reduction: 3, 17, 1549
regulator: 1088.296915546363889203286389249271475784962499
submitted by: David Renshaw
submitted at: 2026-10-21 13:09:18 UTC
last updated: 2026-10-21 13:09:18 UTC

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

(1549, 56538)
(20137/4, 2851705/8)
(196723/9, 87251284/27)
(29091769/18225, 146535626647/2460375)

Commentary

Cube-sum curve x^3 + y^3 = 236997 (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 13:09:19 UTC · history