  Let $\alpha$ be an infinite ordinal and $\gamma$ the unique ordinal
satisfying $\omega^{\omega^\gamma}\leq \alpha < \omega^{\omega^{\gamma+1}}$. We
show that the Banach space $C([0,\,\alpha])$ of all continuous scalar-valued
functions on the compact ordinal interval $[0,\,\alpha]$ has Szlenk index equal
to $\omega^{\gamma+1}$ and $w^\ast$-dentability index equal to
$\omega^{1+\gamma+1}$.
