curve #5804

y² + y = x³ − 5268212791

a-invariants: [0, 0, 1, 0, -5268212791]
c-invariants: [0, 4551735851208]
rank (lower bound): ≥ 4
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 21072851163
discriminant (Δ): -11989756515724662219363
Faltings height: 2.9154
naive height: 58.2931
primes of bad reduction: 3, 7, 13, 307
regulator: 75.37817404100820867916165620718631670152245298
submitted by: jesper-petersen
submitted at: 2026-10-25 10:57:26 UTC
last updated: 2026-10-25 10:57:26 UTC

Witness: 4 independent points log in to add more points →

(3081, 154849)
(3991, 241455)
(6447, 512536)
(83811/25, 22503191/125)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-17,-1,1,16) and specialization t=-1/17. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=27937. PARI/GP 2.17.4 returned matching computational rank bounds [4,4].

last edited by jesper-petersen at 2026-10-25 10:57:27 UTC · history