Computing the Charlap-Coley-Robbins modular polynomials

Abstract

Let $\mathcal{E}$ be an elliptic curve over a field $K$ and $\ell$ a prime. There exists an elliptic curve $\mathcal{E}^*$ related to $\mathcal{E}$ by anisogeny (rational map that is also a group homomorphisms) of degree $\ell$ if and only $\Phi_\ell(X, j(\mathcal{E})) = 0$, where $\Phi_\ell(X, Y)$ is the traditional modular polynomial. Moreover, the modular polynomial gives the coefficients of $\mathcal{E}^*$, together with parameters needed to build the isogeny explicitly. Since the traditional modular polynomial has large coefficients, many families with smaller coefficients can be used instead, as described by Elkies, Atkin and others. In this work, we concentrate on the computation of the family of modular polynomials introduced by Charlap, Coley and Robbins. It has the advantage of giving directly the coefficients of $\mathcal{E}^*$ as roots of these polynomials. We review and adapt the known algorithms to perform the computations of modular polynomials. After describing the use of series computations, we investigate fast algorithms using floating point numbers based on fast numerical evaluation of Eisenstein series. We also explain how to use isogeny volcanoes as an alternative.