I'm currently working on the following exercise from the book [Lectures on Riemann Surfaces, Otto Forster, 1981, page 9]:
Let $\Gamma = \mathbb{Z} \omega_1 + \mathbb{Z} \omega_2$ and $\Gamma' = \mathbb{Z} \omega_1' + \mathbb{Z} \omega_2'$ be two lattices in $\mathbb{C}$. Show that $\Gamma = \Gamma'$ if and only if there exists a matrix $A \in \text{SL}(2,\mathbb{Z})$ such that $\begin{pmatrix} \omega_1' \\ \omega_2' \end{pmatrix} = A \begin{pmatrix} \omega_1 \\ \omega_2 \end{pmatrix}$.
The definition of lattice is as usual, namely $\omega_1,\omega_2$ need to be linearly independent over $\mathbb{R}$. (but there are no assumptions on $\text{Im}(\omega_1/\omega_2)$ or anything)
Now I would claim that the statement in the exercise is false and $\text{SL}(2,\mathbb{Z})$ needs to be replaced with $\text{GL}(2,\mathbb{Z})$. As a counterexample I think of the two lattices spanned by $1,i$ and $1,-i$. They are the same, but the only real matrix $A$ satisfying the above condition is $A = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$, which has determinant $-1$.
Am I right or am I missing something?