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This paper deals with $\tilde{\chi}^{(6)}$, the six-particle contribution tothe magnetic susceptibility of the square lattice Ising model. We havegenerated, modulo a prime, series coefficients for $\tilde{\chi}^{(6)}$. Thelength of the series is sufficient to produce the corresponding Fuchsian lineardifferential equation (modulo a prime). We obtain the Fuchsian lineardifferential equation that annihilates the "depleted" series$\Phi^{(6)}=\tilde{\chi}^{(6)} - {2 \over 3} \tilde{\chi}^{(4)} + {2 \over 45}\tilde{\chi}^{(2)}$. The factorization of the corresponding differentialoperator is performed using a method of factorization modulo a prime introducedin a previous paper. The "depleted" differential operator is shown to have astructure similar to the corresponding operator for $\tilde{\chi}^{(5)}$. Itsplits into factors of smaller orders, with the left-most factor of order sixbeing equivalent to the symmetric fifth power of the linear differentialoperator corresponding to the elliptic integral $E$. The right-most factor hasa direct sum structure, and using series calculated modulo several primes, allthe factors in the direct sum have been reconstructed in exact arithmetics.

High order Fuchsian equations for the square lattice Ising model: $\chi^{(6)}$