curve #5820

y² = x³ − 463901477952675

a-invariants: [0, 0, 0, 0, -463901477952675]
c-invariants: [0, 400810876951111200]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 150304078856666700
discriminant (Δ): -92968379098564122335321131230000
Faltings height: 4.8130
naive height: 81.0645
primes of bad reduction: 2, 3, 5, 7, 31, 73, 157
regulator: 12487.7738627510202615260943201947874041395071090
submitted by: jesper-petersen
submitted at: 2026-10-25 11:01:03 UTC
last updated: 2026-10-25 11:01:03 UTC

Witness: 5 independent points log in to add more points →

(1056139, 1085165788)
(3449761/36, 4405826159/216)
(6750529/4, 17538229633/8)
(6750529/64, 13638555617/512)
(81361639/529, 685504850638/12167)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(13,-8,-5,-1) and specialization t=29/1. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=223833330. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 11:01:03 UTC · history