curve #4733

y² + y = x³ − 1

a-invariants: [0, 0, 1, 0, -1]
c-invariants: [0, 648]
rank (lower bound): ≥ 1
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 243 ☆ record for cube-sum curves, rank ≥ 1
discriminant (Δ): -243 ☆ record for Mordell curves, rank ≥ 1
Faltings height: -0.8634 ☆ record for Mordell curves, rank ≥ 1
naive height: 12.9478 ☆ record for Mordell curves, rank ≥ 1
primes of bad reduction: 3
regulator: 0.5063673553800989597196501202430774640744
submitted by: David Renshaw
submitted at: 2026-10-20 21:27:09 UTC
last updated: 2026-10-20 21:27:09 UTC

Witness: 1 independent point log in to add more points →

(1, 0)

Commentary

Cube-sum curve x^3 + y^3 = 9 (Weierstrass model y^2 = x^3 - 432*9^2). 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 21:27:09 UTC · history