A mapping torus, $M \rtimes_\varphi S^1$, is a fiber bundle over $S^1$ with fiber $M$, where $\varphi$ is an element of mapping class group of $M$, describing the twist around $S^1$. For $M=S^1\times S^1 = T^2$, where the two $S^1$ are parametrized by $x$ and $y$, the map $\varphi$ is given by $$ \begin{pmatrix} x\\ y\\ \end{pmatrix} \to \begin{pmatrix} a & b\\ c & d\\ \end{pmatrix} \begin{pmatrix} x\\ y\\ \end{pmatrix} ,\ \ \ \ a,b,c,d \in \mathbb{Z}, \ ad-bc =\pm 1. $$

What is the cohomology ring of the mapping torus $M \rtimes_\varphi S^1$ in terms of $ \begin{pmatrix} a & b\\ c & d\\ \end{pmatrix}$?

added: I mean to ask the ring $H^∗(M;\mathbb{Z})$ and $H^∗(M;\mathbb{Z}_n)$.