curve #5836

y² = x³ − 2600920724076675

a-invariants: [0, 0, 0, 0, -2600920724076675]
c-invariants: [0, 2247195505602247200]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 3467894298768900
discriminant (Δ): -2922388680786423279421512552030000
Faltings height: 5.1004
naive height: 84.5124
primes of bad reduction: 2, 3, 5, 7, 11, 13, 37, 53
regulator: 19387.6747782836687451957173639050020847682190012
submitted by: jesper-petersen
submitted at: 2026-10-25 11:04:25 UTC
last updated: 2026-10-25 11:04:25 UTC

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

(306219, 161595972)
(313131, 167636196)
(750841/4, 506790739/8)
(2771041/4, 4594719311/8)
(4252521/4, 8759900331/8)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-10,-9,19,-1) and specialization t=6/5. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=19629610. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 11:04:25 UTC · history