Square lattice Ising model χ⁽⁵⁾ ODE in exact arithmetic

We obtain in exact arithmetic the order 24 linear differential operator L₂₄ and the right-hand side E⁽⁵⁾ of the inhomogeneous equation L₂₄(Φ⁽⁵⁾) = E⁽⁵⁾, where φ⁽⁵⁾ = χ̃⁽⁵⁾ - χ̃⁽³⁾/2 + χ̃⁽¹⁾/120 is a linear combination of n-particle contributions to the susceptibility of the square lattice Ising model. In Bostan et al (2009 J. Phys. A: Math. Theor. 42 275209), the operator L₂₄ (modulo a prime) was shown to factorize into L^(left) ₁₂ · L^(right) ₁₂; here we prove that no further factorization of the order 12 operator L^(left) ₁₂ is possible. We use the exact ODE to obtain the behaviour of χ̃ ⁽⁵⁾ at the ferromagnetic critical point and to obtain a limited number of analytic continuations of χ̃⁽⁵⁾ beyond the principal disc defined by its high temperature series. Contrary to a speculation in Boukraa et al (2008 J. Phys. A: Math. Theor. 41 455202), we find that χ̃⁽⁵⁾ is singular at w = 1/2 on an infinite number of branches.