  Let $K$ be a number field and $E_1, \ldots, E_n$ be elliptic curves over $K$,
pairwise non-isogenous over $\overline{K}$ and without complex multiplication
over $\overline{K}$. We study the image of the adelic representation of the
absolute Galois group of $K$ naturally attached to $E_1 \times \cdots \times
E_n$. The main result is an explicit bound for the index of this image in
$\left\{ (x_1,\ldots,x_n) \in \operatorname{GL}_2(\hat{\mathbb{Z}})^n
\bigm\vert \det x_i = \det x_j \;\; \forall i,j \right\}$.
