Let $M_{g,1}$ be the mapping class group of surfaces of genus g $\geq 1$ with one boundary component. By $S_g$ we denote a closed surface of genus $g$.

In the paper "Families of jacobian manifolds and characteristic classes of surface bundles.I" S.Morita proved that the twisted cohomology $H^1(M_{g,1};H^1(S_g;\mathbb{Z}))$ is $\mathbb{Z}$ if $g \geq 2$.

My question is if there is the some result known for $g=1$ that is $H^1(M_{1,1};H^1(S_1;\mathbb{Z}))$ ?