curve #4744

y² = x³ − 5062143495128217077476875

a-invariants: [0, 0, 0, 0, -5062143495128217077476875]
c-invariants: [0, 4373691979790779554940020000]
rank (lower bound): ≥ 8
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 65605379696861693324100300 ☆ record for cube-sum curves, rank ≥ 8
discriminant (Δ): -11070128202596174094225719273559199137491781018750000 ☆ record for cube-sum curves, rank ≥ 8
Faltings height: 8.6652 ☆ record for cube-sum curves, rank ≥ 8
naive height: 127.2908 ☆ record for cube-sum curves, rank ≥ 8
primes of bad reduction: 2, 3, 5, 11, 19, 23, 31, 83, 4201
regulator: 28082310056.0714998395532056377642690916230340384664898
submitted by: David Renshaw
submitted at: 2026-10-20 21:27:49 UTC
last updated: 2026-10-20 21:27:49 UTC

Witness: 8 independent points log in to add more points →

(3510554761849/15876, 4796651982921364243/2000376)
(24718037951641/133956, -54168367502493172061/49027896)
(62260360076731/71289, 489396656251949635504/19034163)
(325400730053089/1089936, -5282120315328637689937/1137893184)
(468784925321359/2556801, 4290589752596243996102/4088324799)
(768132040232539/4173849, -9226941692672344607288/8527173507)
(2847716696637739/1279161, -151930694548929806737388/1446731091)
(5709567766269799/4272489, -430966746133174586175218/8831234763)

Commentary

Cube-sum curve x^3 + y^3 = 23381862574950 (Weierstrass model y^2 = x^3 - 432*23381862574950^2). 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 21:27:49 UTC · history