  Let $\mathcal{E}$ be an elliptic curve over a field $\mathbf{K}$ and $\ell$ a
prime. There exists an elliptic curve $\mathcal{E}^*$ related to $\mathcal{E}$
by an isogeny of degree $\ell$ only if $\Phi_\ell^t(X, j(\mathcal{E})) = 0$,
where $\Phi_\ell^t(X, Y)$ is the traditional modular polynomial. Moreover,
$\Phi_\ell^t$ gives the coefficients of $\mathcal{E}^*$, together with
parameters needed to build the isogeny explicitly. Since $\Phi_\ell^t$ has very
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 Fricke
and more recently used by Charlap, Coley and Robbins. In some cases, the
resulting polynomials are small, which justifies the interest of this study. We
review and adapt the known algorithms to perform the computations of these
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. The last part is concerned with finding explicit formulas for
computing the coefficients of $\mathcal{E}^*$. To this we add tables of
numerical examples.
