We present two algorithms that, given a prime
$\ell $ and an elliptic curve
$E\u2215{\mathbb{F}}_{q}$, directly compute the
polynomial
${\Phi}_{\ell}\left(j\left(E\right),Y\right)\in {\mathbb{F}}_{q}\left[Y\right]$ whose roots are
the
$j$invariants of the elliptic
curves that are
$\ell $isogenous
to $E$.
We do not assume that the modular polynomial
${\Phi}_{\ell}\left(X,Y\right)$ is
given. The algorithms may be adapted to handle other types of modular
polynomials, and we consider applications to point counting and the
computation of endomorphism rings. We demonstrate the practical efficiency
of the algorithms by setting a new pointcounting record, modulo a prime
$q$ with
more than 5,000 decimal digits, and by evaluating a modular polynomial of level
$\ell =\text{100,019}$.
