We present an unconditional CRT algorithm to compute the modular polynomial $\Phi_\ell(X,Y)$ in quasi-linear time. The main ingredients of our algorithm are: the embedding of $\ell$-isogenies in smooth-degree isogenies in higher dimension, and the computation of m-th order deformations of isogenies. We provide a proof-of-concept implementation of a heuristic version of the algorithm demonstrating the practicality of our approach. Our algorithm can also be used to compute the reduction of $\Phi_\ell$ modulo $p$ in quasi-linear time (with respect to $\ell$) $\tilde{O}(\ell^2 ( \log p + \log \ell)^\mathfrak{O})$.