On l-th roots and division by l

We give a characterization of the codomain []E(k) of the multiplication-by- map [] in the case of elliptic curves over a field k of characteristic ≠ 2,3 with -torsion E[]= W₁,W₂ fully defined over k, for primes different from the characteristic. We show that a point Q∈ E(k) lies in []E(k) if and only if hW₁(-Q) and hW₂(-Q) are -powers of k, where hW₁ and hW₂ are functions on E with divisor div(hWᵢ)= Wᵢ- P∞. Our characterization leads to an effective procedure to find pre-images of [] by solving an order system of linear equations and computing a polynomial gcd.