Mistaken in Silverman's AAEC Theorem 4.3 ? Where am I mistaken?

Question

According to Silverman's ''Advanced topics in the arithmetic of elliptic curves'', p122, theorem 4.3,

Theorem4.3: Let $E_1,E_2,・・・,E_h$ be complete set of representatives for $ell(R_K)$(elliptic curves which has complex multiplication over $K$, up to isomorphism over $ \Bbb{C}$), then $j(E_1),j(E_2),・・・,j(E_h)$ is a complex set of $Gal( \overline{K} /{K}$) conjugates for $j(E)$.

But $ell( \Bbb{Z}[ \sqrt{-5} )$ consist of two elements $E_1,E_2$, and $j(E_1)=632000+282880\sqrt{-5}$, $j(E_2)=632000-282880\sqrt{-5}$.

In this situation, $K= \Bbb{Q}( \sqrt{-5})$. So $Gal( \overline{K} /{K}$) conjugates for $j(E_1)$ is $j(E_1)$. This is contradiction because $j(E_1)≠j(E_2)$.

So Theorem 4.3 seems to be false in this case. Where am I mistaken ?

Answer 1

Your $632000+282880\sqrt{-5}$ isn't the $j$-invariant of an elliptic curve with CM by $O_K=\Bbb{Z}[\sqrt{-5}]$.

This is because the lattices with CM by $O_K$ are of the form $\alpha (\Bbb{Z}+\sqrt{-5}\Bbb{Z})$ and $\alpha (2\Bbb{Z}+(1+\sqrt{-5})\Bbb{Z})$.

In both cases their $j$-invariant $j(\sqrt{-5})$ and $j(\frac{1+\sqrt{-5}}2)$ are real.

(recall that $j(z)=\sum_{n\ge -1} c_n e^{2i\pi n z}$ with $c_n\in \Bbb{R}$)