Summary: We consider the Fuchsian linear differential equation obtained (modulo a prime) for \(\widetilde{\chi}^{(5)}\), 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 \(\widetilde{\chi}^{(1)}\) and \(\widetilde{\chi}^{(3)}\) can be removed from \(\widetilde{\chi}^{(5)}\) 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 \(\widetilde{\chi}^{(3)}\) and \(\widetilde{\chi}^{(4)}\). 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 \(\widetilde{\chi}^{(n)}\)’s.