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?

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 $\mathbf{Q}$.

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?

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 $\mathbf{Q}$.

Let $E$ be an elliptic curve over $\mathbf{C}$. Then if we apply the uniformization theorem, we can write $E(\mathbf{C})$ as follows: $$E(\mathbf{C}) \cong \mathbf{C}/(\mathbf{Z} + \mathbf{Z} \tau ) $$ 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?

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?

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 $\mathbf{Q}$.