curve #4745

y² = x³ − 19123867603931312593446075

a-invariants: [0, 0, 0, 0, -19123867603931312593446075]
c-invariants: [0, 16523021609796654080737408800]
rank (lower bound): ≥ 8
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 76495470415725250373784300
discriminant (Δ): -157992038841323619249698570739866368205346979015230000
Faltings height: 8.8867
naive height: 129.9491
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 29, 41, 47, 59
regulator: 2145812661.93237672769405123774575293469121536119078234
submitted by: David Renshaw
submitted at: 2026-10-20 21:27:53 UTC
last updated: 2026-10-20 21:27:53 UTC

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

(269938579, 738716913908)
(283984639, 1943892930862)
(644782051, 15777843090976)
(1679172499, 68669534486132)
(1502372041/4, -46552218884261/8)
(2186568211, 102152018376316)
(5468797489/4, 402908772015287/8)
(1196456469121/144, 1308694006383930431/1728)

Commentary

Cube-sum curve x^3 + y^3 = 1683200989470 (Weierstrass model y^2 = x^3 - 432*1683200989470^2). 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 21:27:53 UTC · history