Which lens spaces are chiral?

Question

By chiral, I mean a manifold which does not admit a orientation-reversing self-homeomorphism.

I am curious if the lens space $L(p,q)$ is chiral or not, for varying values of $p$ and $q$. For instance, is $L(3,1)$ chiral?

Answer 1

Example 3.22 and Lemma 3.23 in Hempel give $q^2\equiv -1\pmod{p}$ as a necessary and sufficient condition for $L(p, q)$ to admit an orientation-reversing homeomorphism. The proof's not particularly difficult but does require a bit of 3-manifold topology (and if you're already familiar with 3-manifold topology, I don't have anything to add to Hempel's proof).