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}$?