curve #4734

y² = x³ − 243

a-invariants: [0, 0, 0, 0, -243]
c-invariants: [0, 209952]
rank (lower bound): ≥ 1
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 972
discriminant (Δ): -25509168
Faltings height: 0.1001
naive height: 24.5093
primes of bad reduction: 2, 3
regulator: 2.444086321700477944323091171694727367779
submitted by: David Renshaw
submitted at: 2026-10-20 21:27:13 UTC
last updated: 2026-10-20 21:27:13 UTC

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

(7, 10)

Commentary

Cube-sum curve x^3 + y^3 = 6 (Weierstrass model y^2 = x^3 - 432*6^2). 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 21:27:13 UTC · history