  We compute the nonabelian $\mathrm{SL_2}(\mathbb{C})$-character varieties of
the rational knots $C(2n+1,2m,2)$ in the Conway notation, where $m$ and $n$ are
non-zero integers. By studying real points on these varieties, we determine the
left orderability of the fundamental groups of the cyclic branched covers of
$C(2n+1,2m,2)$.
