 Solution

Choose essential curves $\alpha ,\beta$ on $T^2$ with $i(\alpha ,\beta )=1$. They fill the torus. Let $[f]$ be central in $\mathrm{Mod}(T^2)$. For every curve $\gamma$,We also use that equality of Dehn twists about essential curves implies equality of their unoriented isotopy classes. Since $[f]$ commutes with $T_{\alpha }$ and $T_{\beta }$, it preserves both $\alpha$ and $\beta$.

The structure graph of this filling pair has one intersection vertex in the embedded union. Its orientation-preserving symmetries induced by a torus homeomorphism are the identity and simultaneous reversal of both curves. By the Alexander method, the corresponding mapping classes are the identity and the elliptic involutionThe involution commutes with every torus mapping class, as is also clear from the identification $\mathrm{Mod}(T^2)\cong S{L}_2(\mathbb{Z})$ where it is $-I$. Therefore the center of the mapping class group of the torus is