curve #5799

y² = x³ − 621075

a-invariants: [0, 0, 0, 0, -621075]
c-invariants: [0, 536608800]
rank (lower bound): ≥ 3
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 201228300
discriminant (Δ): -166637155230000
Faltings height: 1.4078
naive height: 40.2016
primes of bad reduction: 2, 3, 5, 7, 13
regulator: 41.913963818865039378260571348651601785800765
submitted by: jesper-petersen
submitted at: 2026-10-25 10:54:58 UTC
last updated: 2026-10-25 10:54:58 UTC

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

(91, 364)
(1729/4, 71617/8)
(1729/16, 51233/64)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-22,-1,3,19) and specialization t=-17/12. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=8190. PARI/GP 2.17.4 returned matching computational rank bounds [3,3].

last edited by jesper-petersen at 2026-10-25 10:54:58 UTC · history