lcm in $\mathbb{Z}[\sqrt{-5}]$ does not exists 
I need to show that lcm of $2$ and $1+\sqrt{-5}$ does not exists in $\mathbb{Z}[\sqrt{-5}]$

Getting no idea about how to start, I was thinking when does lcm cannot exists!

Let $a=x+y\sqrt{-5}$ be an LCM of $2$ and $1+\sqrt{-5}$. Then $(a)$ is equal to the ideal $I:=(2) \cap (1+\sqrt{-5})$. It can be seen that $I$ has index $12$ in $\mathbb{Z}[\sqrt{-5}]$. But $(a)$ has index $Nm(a)=x^2+5y^2$ which is different from $12$.
