  We provide an easy method for the construction of characteristic polynomials
of simple ordinary abelian varieties ${\mathcal A}$ of dimension $g$ over a
finite field ${\mathbb F}_q$, when $q\ge 4$ and $2g=\rho^{b-1}(\rho-1)$ for
some prime $\rho\ge 5$, with $b\ge 1$. Moreover, we show that ${\mathcal A}$ is
absolutely simple if $b=1$ and $\rho=5$, but ${\mathcal A}$ is not absolutely
simple for arbitrary $\rho$ with $b>1$.
