  The degree of irrationality of a smooth projective variety $X$ is the minimal
degree of a dominant rational map $X\dashrightarrow \mathbb{P}^{\dim X}$. We
show that if an abelian surface $A$ over $\mathbb{C}$ is such that the image of
the intersection pairing $\text{Sym}^2NS(A)\to \mathbb{Z}$ does not contain
$12$, then it has degree of irrationality $4$. In particular, a very general
$(1,d)$-polarized abelian surface has degree of irrationality $4$ provided that
$d\nmid 6$. This answers two questions of Yoshihara by providing the first
examples of abelian surfaces with degree of irrationality greater than $3$ and
showing that the degree of irrationality is not isogeny-invariant for abelian
surfaces.
