Integral cohomology of a circle bundle over a product of two spheres

Let $a,b$ be the standard basis of $H^2(S^2\times S^2;\mathbb{Z})$ and let $E\to S^2\times S^2$ be an oriented circle bundle with Euler class $pa+qb\ne 0$. If $d=\gcd (|p|,|q|)$, its Gysin sequence givesFor zero Euler class the bundle is trivial and its cohomology is that of $(S^2\times S^2)\times S^1$.