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?
