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As a subproduct of the Schoof-Elkies-Atkin algorithm to count points on elliptic curves defined over finite fields of characteristic $p$, there exists an algorithm that computes, for $\ell$ an Elkies prime, $\ell$-torsion points in an extension of degree $\ell -1$ at cost $\tilde{O}(\ell max{(\ell ,logq)}^2)$ bit operations in the favorable case where $\ell \le p/2$. We combine in this work a fast algorithm for computing isogenies due to Bostan, Morain, Salvy and Schost with the $p$-adic approach followed by Joux and Lercier to get an algorithm valid without...