I know that you can take any element from $R = \mathbf{Z}[\sqrt{-5}]$ like, say, $a = 1 + \sqrt{-5}$, and it will be a principal ideal, but when I used the norms I got this:
$N(a) = N(1 + \sqrt{-5})N(1) = 6,$ so $N(a)$ must be either 1, 2 or 3. $N(a)$ can't be 1 because then $(a) = (1+\sqrt{-5})$ wouldn't be a proper ideal anymore. All that's left is to find out if the equations $a^2 + 5b^2 = 2$ and $a^2 + 5b^2 = 3$ have any solutions. If either one of these has solutions then it confirms that $(1 + \sqrt{-5})$ is a principal ideal, so my question is, are there any?