curve #4736

y² + y = x³ − 3997

a-invariants: [0, 0, 1, 0, -3997]
c-invariants: [0, 3453192]
rank (lower bound): ≥ 3
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 1294947 ☆ record for cube-sum curves, rank ≥ 3
discriminant (Δ): -6900772563 ☆ record for cube-sum curves, rank ≥ 3
Faltings height: 0.5668 ☆ record for cube-sum curves, rank ≥ 3
naive height: 30.1096 ☆ record for cube-sum curves, rank ≥ 3
primes of bad reduction: 3, 73
regulator: 13.805737597671224689075985016522916911803109
submitted by: David Renshaw
submitted at: 2026-10-20 21:27:21 UTC
last updated: 2026-10-20 21:27:21 UTC

Witness: 3 independent points log in to add more points →

(73/4, 361/8)
(73, 620)
(3577/9, 213913/27)

Commentary

Cube-sum curve x^3 + y^3 = 657 (Weierstrass model y^2 = x^3 - 432*657^2). 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.

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