Degree of irrationality of abelian surfaces

Let $A$ be a complex abelian surface. We prove that $A$ admits a dominant rational map of degree three to ${\text{PP}}^2$ if and only if it admits a polarization of type $(1,1)$, $(1,2)$, $(1,3)$, or $(1,6)$. Consequently, the degree of irrationality of $A$ is $3$ precisely in these four cases and is $4$ otherwise.