  In [3], it is proved that the quotient of an abelian variety $A$ by a finite
order automorphism $g$ is uniruled if and only if some power of $g$ satisfies a
numerical condition $0<\age(g^k)<1$. In this paper, we show that $\age(g^k)=1$
is enough to guarantee that $A/\langle g\rangle$ has at least one rational
curve.
