curve #5800

y² + y = x³ − 592741

a-invariants: [0, 0, 1, 0, -592741]
c-invariants: [0, 512128008]
rank (lower bound): ≥ 3
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 192048003
discriminant (Δ): -151779569778963
Faltings height: 1.4000
naive height: 40.1082
primes of bad reduction: 3, 7, 127
regulator: 29.629967505526871749207409459031538546977943
submitted by: jesper-petersen
submitted at: 2026-10-25 10:55:12 UTC
last updated: 2026-10-25 10:55:12 UTC

Witness: 3 independent points log in to add more points →

(127, 1206)
(889/4, 25777/8)
(118237/81, 40652272/729)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(20,-20,0,-1) and specialization t=0/1. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=8001. PARI/GP 2.17.4 returned matching computational rank bounds [3,3].

last edited by jesper-petersen at 2026-10-25 10:55:12 UTC · history