curve #5824

y² + y = x³ − 252809461319221

a-invariants: [0, 0, 1, 0, -252809461319221]
c-invariants: [0, 218427374579806728]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 81910265467427523
discriminant (Δ): -27610253452446295361875223283603
Faltings height: 4.7119
naive height: 79.8505
primes of bad reduction: 3, 7, 19, 31, 61, 73
regulator: 2246.14854167732547611736400018040405779973602941
submitted by: jesper-petersen
submitted at: 2026-10-25 11:01:43 UTC
last updated: 2026-10-25 11:01:43 UTC

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

(83731, 18281645)
(592249/4, 437672007/8)
(251503, 125122742)
(592249/9, 153096353/27)
(2622817/36, 2499544493/216)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-8,-1,2,6) and specialization t=-22/17. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=165237471. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 11:01:43 UTC · history