curve #4743

y² + y = x³ − 661904741917321

a-invariants: [0, 0, 1, 0, -661904741917321]
c-invariants: [0, 571885697016565128]
rank (lower bound): ≥ 6
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 2647618967669283
discriminant (Δ): -189266927344978315129990066820403
Faltings height: 4.8723
naive height: 81.7754
primes of bad reduction: 3, 73, 103, 439
regulator: 28924832.679234045110111284402075646699867742541409
submitted by: David Renshaw
submitted at: 2026-10-20 21:27:45 UTC
last updated: 2026-10-20 21:27:45 UTC

Witness: 6 independent points log in to add more points →

(634666753/5776, 11317948775967/438976)
(1023719977/484, 32753414198001/10648)
(1812172537/10404, 72150583246111/1061208)
(17094269353/179776, -1072077799685397/76225024)
(10704126731473/15968016, 34982367236825796487/63808191936)
(236991605579281/1784217600, 3090479185676774967929/75365351424000)

Commentary

Cube-sum curve x^3 + y^3 = 9902523 (Weierstrass model y^2 = x^3 - 432*9902523^2). Source: Elkies & Rogers, 'Elliptic Curves x^3+y^3=k of High Rank', ANTS-VI (LNCS 3076, 2004), arXiv:math/0403116, https://arxiv.org/abs/math/0403116 Points mapped from the 3-isogenous curve uv(u+v)=A given there via the dual isogeny.

last edited by David Renshaw at 2026-10-20 21:27:45 UTC · history