Galois extensions of Q(T): first examples

Let E be a finite Galois extension of Q(T ) with group G which is regular, i.e., Q¯ ∩ E = Q. Geometrically, E can be viewed as the function field of a smooth projective curve C which is absolutely irreducible over Q; the inclusion Q(T ) ↪→ E corresponds to a (ramified) Galois covering C −→ P1 defined over Q with group G.