  We study tori which are cyclic covers of the standard torus, that is, the
deck transformation group of the covering map is cyclic. These covering tori
can be parametrized in a natural way and we show that being cyclic is
equivalent to certain arithmetic condition on these parameters. There is a
natural $\mathrm{SL}(2,\mathbb{Z})$-action on covering tori and introducing a
complete numeric $\mathrm{SL}(2,\mathbb{Z})$-invariant we show that, for
$n\in\mathbb{N}$, all $n$-tuple cyclic covers are in the same
$\mathrm{SL}(2,\mathbb{Z})$-orbit. We show that cyclic covers are irreducible
in a precise sense and we give the exact and asymptotic number of these covers.
