the computation of the hilbert class polynomial has applications ranging from explicit class field theory to cryptography. several new algorithms to compute it have been developed during the last 5 years, each having its pro's and cons. in this talk we will present a significant speed up of the `chinese remainder theorem approach'. we will give a detailed run time analysis of the new algorithm, using tools from both analytic number theory and arithmetic geometry. the resulting run time is almost optimal: one of the bottlenecks is writing down the answer.