Let $E/ \Bbb{C} $ be an elliptic curve which has CM over $\Bbb{Q}( \sqrt{-5})$.
Then, why $j(E)$ is real number?
If theory of complex multiplication is well known, we can explicitly calculate $j$ invariant of $E$, but ''Advanced topics in the arithmetic of elliptic curves'' written by Silverman reads $j(E)$ is real number is easy in this situation.
What is a easy way to prove $j($)$ is real number in this situation ?
Best Answer
"has CM over $\Bbb{Q}(\sqrt{-5})$" is ambiguous please don't use it. An elliptic can have CM by $\Bbb{Z}[n\sqrt{-5}]$ but not by $\Bbb{Z}[\sqrt{-5}]$ (try $E \cong \Bbb{C}/\Bbb{Z}[7\sqrt{-5}]$)
If $E$ has CM by $O_K$ then $E$ is isomorphic to $\Bbb{C}/I$ where $I$ is an ideal of $O_K$.
$\overline{I} = \{ a \in O_K,\overline{a}\in I\}$ is an ideal as well.
Then $j(E)$ is real iff $j(\Bbb{C}/\overline{I})=j(\Bbb{C}/I)$ iff $\Bbb{C}/\overline{I} \cong \Bbb{C}/I$ iff $\overline{I}$ is in the same ideal class as $I$.