  Following an idea of Ishida, we develop polynomial equations for certain
unramified double covers of surfaces with p_g=q=1 and K^2=2. Our first main
result provides an explicit surface surface X with these invariants defined
over Q that has Picard number 2, which is the smallest possible for these
surfaces. This is done by giving equations for the double cover Y of X,
calculating the zeta function of the reduction of Y to F_3, and extracting from
this the zeta function of the reduction of X to F_3; the basic idea used in
this process may also be of independent interest.
  Our second main result is a big monodromy theorem for a family that contains
all surfaces with p_g=q=1, K^2=2, and K is ample. It follows from this that a
certain Hodge correspondence of Kuga and Satake, between such a surface and an
abelian variety, is motivated (and hence absolute Hodge). This allows us to
deduce our third main result, which is that the Tate Conjecture in
characteristic zero holds for all surfaces with p_g=q=1, K^2=2, and K ample.
