curve #5819

y² = x³ − 787574198514675

a-invariants: [0, 0, 0, 0, -787574198514675]
c-invariants: [0, 680464107516679200]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 255174040318754700
discriminant (Δ): -267957987047726128549714233630000
Faltings height: 4.9012
naive height: 82.1231
primes of bad reduction: 2, 3, 5, 11, 13, 17, 31, 43
regulator: 45225.8394539718495042643015059165156802777063102
submitted by: jesper-petersen
submitted at: 2026-10-25 11:00:49 UTC
last updated: 2026-10-25 11:00:49 UTC

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

(94471, 7453706)
(201799, 86198918)
(235339, 110664112)
(2080039, 2999771762)
(24746371/25, 123052623044/125)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-6,-1,10,-4) and specialization t=-2/23. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=291647070. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 11:00:49 UTC · history