I am reading Chapter 11 of Dale Husemöller's * Elliptic Curves* Springer book and I got stuck on Theorem (1.4) (c.f., image below).

* Notation and definitions:* Let $L$ and $L'$ be two complex lattices spanned, respectively, by periods $\{\psi_1,\psi_2\}$ and $\{\psi_1',\psi_2'\}$. Set $\tau:=\psi_2/\psi_1\in\mathfrak{H}$ and $\tau':=\psi_2'/\psi_1'\in\mathfrak{H}$, where $\mathfrak{H}$ is the upper-half complex plane. Finally, we introduce two tori $T_\tau:=\mathbb{C}/L$ and $T_{\tau'}:=\mathbb{C}/L'$.

By definition, $T_{\tau}$ and $T_{\tau'}$ are said to be isogenic iff there's a complex number $\lambda$ such that $\lambda L'\subset L$.

Below, $GL^+_2(\mathbb{K})$ denotes the group of two-by-two invertible matrices with positive determinant and with entries in the field $\mathbb{K}$.

* My question regarding the proof of (2) below is the following:* Why do we need to restrict to the subgroup $GL^+_2(\mathbb{Q})$ of $GL^+_2(\mathbb{R})$ and not just take $GL^+_2(\mathbb{R})$ itself? I see no obstruction in replacing $\mathbb{Q}$ by $\mathbb{R}$ in the proof.

Am I missing something here?