  Let K be a number field and let $\mathcal{E}$ be an elliptic curve defined
over $K$. Let $m$ be a positive integer. We denote by $K(\mathcal{E}[m])$ the
number fields obtained by adding to $K$ the coordinates of the $m$-torsion
points of $\mathcal{E}$. We look for small (sometimes "minimal") set of
generators of $K(\mathcal{E}[m])$. For $m=3$ and $m=4$, we describe explicit
generators, degree and Galois groups of the extensions $K(\mathcal{E}[m])/K$.
