K-theory of the mapping torus of the factor swap on two complex projective planes ($K^0(T_{\mathrm{swap}})$)

For the factor swap on ${\mathbb{CP}}^2\times {\mathbb{CP}}^2$, the invariant subgroup ofhas basis $1,xy,x^2{y}^2,x+y,x^2+y^2,x{y}^2+x^{2}y$. Hence the mapping torus has $K^0\cong {\mathbb{Z}}^6$.