curve #5325

y² = x³ − 571485612

a-invariants: [0, 0, 0, 0, -571485612]
c-invariants: [0, 493763568768]
rank (lower bound): ≥ 4
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 46290334572
discriminant (Δ): -141089387640342283008
Faltings height: 2.5452
naive height: 53.8506
primes of bad reduction: 2, 3, 67, 103
regulator: 551.3322247734175678025697854399324257517050915
submitted by: David Renshaw
submitted at: 2026-10-21 13:09:23 UTC
last updated: 2026-10-21 13:09:23 UTC

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

(2077, 91589)
(6901, 572783)
(21733, 3203815)
(20569/4, 2943779/8)

Commentary

Cube-sum curve x^3 + y^3 = 248436 (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 13:09:23 UTC · history