  The aim of the inverse Galois problem is to find extensions of a given field
whose Galois group is isomorphic to a given group. In this article, we are
interested in subgroups of GL(2,Z/nZ) where n is an integer. We know that, in
general, we can realize these groups as the Galois group of a given number
field, using the torsion points on an elliptic curve. Specifically, a theorem
of Reverter and Vila (2000) gives, for each prime n, a polynomial, depending on
an elliptic curve, whose Galois group is GL(2,Z/nZ). In this article, we
generalize this theorem in several directions, in particular for n non
necessarily prime. We also determine a minimum for the valuations of the
coefficients of the polynomials arising in our construction, depending only on
n.
