  We consider the Fuchsian linear differential equation obtained (modulo a
prime) for $\tilde{\chi}^{(5)}$, the five-particle contribution to the
susceptibility of the square lattice Ising model. We show that one can
understand the factorization of the corresponding linear differential operator
from calculations using just a single prime. A particular linear combination of
$\tilde{\chi}^{(1)}$ and $\tilde{\chi}^{(3)}$ can be removed from
$\tilde{\chi}^{(5)}$ and the resulting series is annihilated by a high order
globally nilpotent linear ODE. The corresponding (minimal order) linear
differential operator, of order 29, splits into factors of small orders. A
fifth order linear differential operator occurs as the left-most factor of the
"depleted" differential operator and it is shown to be equivalent to the
symmetric fourth power of $L_E$, the linear differential operator corresponding
to the elliptic integral $E$. This result generalizes what we have found for
the lower order terms $\tilde{\chi}^{(3)}$ and $\tilde{\chi}^{(4)}$. We
conjecture that a linear differential operator equivalent to a symmetric
$(n-1)$-th power of $L_E$ occurs as a left-most factor in the minimal order
linear differential operators for all $\tilde{\chi}^{(n)}$'s.
