Let $E$ be an elliptic curve over $\mathbf{Q}$. Then we can base-change $E$ to $\mathbf{C}$ and apply the uniformization theorem to obtain:
$$E(\mathbf{C}) \cong \mathbf{C}/(\mathbf{Z} + \mathbf{Z} \tau ) $$
for some complex number $\tau$ in the upper half plane. I've done a few numerical tests on Sage, and I've found that the *real part* of $\tau$ seems to be a *rational number*. So I wanted to ask: is it known to be true that $\rm{Re }\,\,\tau$ is a rational number, and if it is, would anyone be able to provide a reference for a proof?   

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*Important note*: an earlier version of the question said that $E$ was an elliptic curve defined over $\mathbf{C}$, *not* over $\mathbf{Q}$, which explains some of the comments. I've just edited it to say that $E$ must be defined over $\mathbf{Q}$.