Study and computation of a Hurwitz space and totally real PSL2(F8)-extensions of Q

Developing on works by Fried, Völklein, Matzat, Malle, Dèbes, Wewers, we give a method for computing a Hurwitz space and illustrate it on some example of number theoretic interest: we study and compute a family of degree 9 covers of PC1 with monodromy group PSL2(F8) and having four branch points. We deduce explicit regular PSL2(F8)-extensions of the rational function field Q(φ) with totally real fibers. This gives rise to totally real polynomials over Q with Galois group PSL2(F8).