  We present an algorithm that, on input of a CM-field $K$, an integer $k\ge1$,
and a prime $r \equiv 1 \bmod k$, constructs a $q$-Weil number $\pi \in \O_K$
corresponding to an ordinary, simple abelian variety $A$ over the field $\F$ of
$q$ elements that has an $\F$-rational point of order $r$ and embedding degree
$k$ with respect to $r$. We then discuss how CM-methods over $K$ can be used to
explicitly construct $A$.
