Let $M=\mathbb{(Z[\sqrt{-5} ])^2/ \begin{bmatrix} 2 \\ 1+\sqrt{-5} \end{bmatrix} Z[\sqrt{-5}]}$, a quotient $\mathbb{Z}[\sqrt{-5}]$-module. Can we find a basis $B$ of $M$ such that $|B|=1$?
I believe that the answer is "no," but I am having trouble proving this.
If I start by assuming $B=\{\begin{bmatrix} a \\ b \end{bmatrix}+\begin{bmatrix} 2 \\ 1+\sqrt{-5} \end{bmatrix} \mathbb{Z}[\sqrt{-5}]\}$ is a basis for $M$, it is not clear to me how I may obtain a contradiction from this. Please help.