An explicit open image theorem for products of elliptic curves

Let K be a number field and let E₁,…,En be elliptic curves over K, pairwise non-isogenous over K̅ and without complex multiplication over K̅. We study the image G∞ of the adelic representation of Gal (K̅/K) naturally attached to E₁ × … × En. The main result is an explicit bound for the index of G∞ in {( x₁,…,xn)∈ GL₂ (ℤ̂)ⁿ | det xi = det xj ∀i,j}