Let $p$ and $q$ be co-prime integers , $p \neq 0$. Let $r$ denote the rotation of $\mathbb{R}^3$ , with a vertical axis oriented along the canonical basis, and with an angle of $2\pi / p$, and denote by $\sigma$ the orthogonal symmetry with respect to the plane $x_3 = 0$. The lents space $L_{p,q}$ is the quotient of the ball $ \mathbb{B}^3$ by the equivalence relation $R$ which identifies a point $x$ of the northern hemisphere $S_{+}^{2}$ of the edge $S^2$ of $\mathbb{B}^3$ with the point $\sigma(r^q(x))$ of the southern hemisphere $S_{-}^{2}$ from this same side.
(a) What space do we get if $p = 1$? What if $p = 2$?
I could find something related to lens spaces here but I am not sure what $q$ is, in my homework problem. Is $q$ known here?
Moreover, the fundamental group of lens spaces as proved here seems to be independent of $q$ in my case. It is simply isomorphic to $\mathbb{Z}_p$.