curve #5832

y² = x³ − 22815637920075

a-invariants: [0, 0, 0, 0, -22815637920075]
c-invariants: [0, 19712711162944800]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 7392266686104300
discriminant (Δ): -224879040158384565542018430000
Faltings height: 4.3110
naive height: 75.0401
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 29
regulator: 20284.7986838920590929026992547982165479859000831
submitted by: jesper-petersen
submitted at: 2026-10-25 11:03:32 UTC
last updated: 2026-10-25 11:03:32 UTC

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

(32851, 3554824)
(136591, 50255114)
(707161/4, 593442941/8)
(2159521/4, 3173252719/8)
(5240599/121, 10173882082/1331)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-7,-3,-1,10) and specialization t=-6/17. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=49639590. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 11:03:32 UTC · history