Polynomials realizing images of Galois representations of an elliptic curve

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 GL₂ (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 gives, for each prime n, a polynomial, depending on an elliptic curve, whose Galois group is GL₂ (Z/nZ). In this article, we generalize this theorem in several directions, in particular for n not necessarily prime. We also determine a minimum for the valuations of the coefficients of the polynomials arising in our construction, depending only on n.