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We study smooth curves on which the alternating group $\mathfrak{A}_{6}$ actsfaithfully. Let $\mathcal{V} \subset PGL(3, \mathbb{C})$ be the Valentinergroup, which is isomorphic to $\mathfrak{A}_{6}$. We see that there areintegral $\mathcal{V}$-invariant curves of degree $12$ which have geometricgenera $10$ and $19$. On the other hand, if $\mathfrak{A}_{6}$ acts faithfullyon a curve $C$ of genus $10$ or $19$, then we give an explicit description ofthe extension $k(C / \mathfrak{A}_{5}) \rightarrow k(C / \mathfrak{A}_{6})$ forany icosahedral subgroup $\mathfrak{A}_{5}$. Using this, we show the uniquenessof smooth projective curves of genera $10$ and $19$ whose automorphism groupscontain $\mathfrak{A}_{6}$.

A_6 invariant curves of genera 10 and 19