On l-th roots and division by l

We give a characterization of the codomain $[\mathrm{\ell }]E(k)$ of the multiplication-by-$\mathrm{\ell }$ map $[\mathrm{\ell }]$ in the case of elliptic curves over a field $k$ of characteristic $\mathrm{\ne }2,3$ with $\mathrm{\ell }$-torsion $E[\mathrm{\ell }]=⟨W_1,W_2⟩$ fully defined over $k$, for primes $\mathrm{\ell }$ different from the characteristic. We show that a point $Q\in E(k)$ lies in $[\mathrm{\ell }]E(k)$ if and only if $h_{W_1}(-Q)$ and $h_{W_2}(-Q)$ are $\mathrm{\ell }$-powers of $k$, where $h_{W_1}$ and $h_{W_2}$ are functions on $E$ with divisor $\mathrm{d}\mathrm{i}\mathrm{v}(h_{W_i})=\mathrm{\ell }{W}_i-\mathrm{\ell }{P}_{\mathrm{\infty }}$. Our characterization leads to an effective procedure to find pre-images of $[\mathrm{\ell }]$ by solving an order $\mathrm{\ell }$ system of linear equations and computing a polynomial gcd.