Let $E=\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})$ be an alliptic curve. Define a holomorphic 1-form $dz=dx+idy$ and a Kahler form $\omega=dx\wedge dy$. How can one prove that special Lagrangians on $E$ are all given by a line?

By special Lagrangian, I mean a submanifold $L$ of $E$ with $\omega|_L=0$ and $Im(dz)|_L=C \cdot\mathrm{vol}$ for some constant $C$. $\mathrm{vol}$ is the Riemannian volume form induced by the metric associated to $\omega$.