Complex algebraic surfaces of general type with $p_g=q=1$ are still not completely understood. Until recently only a few examples were known. In this talk I will give some results about surfaces $S$ with $p_g=q=1$ having an involution (automorphism of order $2$).

If the bicanonical map $\phi_2$ of $S$ is of degree 2 onto a non-ruled surface, then $\phi_2(S)$ is birational to a $K3$ surface. We will use the Computational Algebra System Magma to construct such a surface with $K_S^2=6,$ as a double cover of a Kummer quartic surface (quartic in $P^3$ with 16 ordinary double points).