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 if $E$ is defined over $\mathbf{Q}$ and $\tau$ is given as above, then $\rm{Re }\,\,\tau$ 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 $\mathbf{Q}$.



The real part of the period of an elliptic curve
However, if $E$ is defined over $\mathbb R$, then it's always possible to find a $\tau$ of the form either $\tau=ti$ or $\tau=\frac12+ti$ so that $E(\mathbb C)$ is analytically isomorphic (over $\mathbb R$, even) to $\mathbb C/(\mathbb Z+\mathbb Z\tau)$. So possibly the examples you were looking at are defined over $\mathbb R$, which you can check by seeing if $j(E)\in\mathbb R$.


**Addendum** If $E$ has complex multiplication, then $\mathbb Q(\tau)$ is an imaginary quadratic field. If $E$ does not have CM, then my recollection is that $\tau$ is transcendental over $\mathbb Q$. There is further information in the answer to [When is the period of elliptic curve over the rationals transcendental?](https://mathoverflow.net/questions/28385/when-is-the-period-of-elliptic-curve-over-the-rationals-transcendental)


