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It is well-known that every CM elliptic curve has $j$-invariants which are algebraic integers. Is there any criterion to classify all the $j$-invariant that corresponds to $CM$ elliptic curves? For small degrees, such as over $\mathbb{Q}$, we can do it by hand using modular polynomials. Also, we can do this for any algebraic integer with an arbitrarily large degree since the number of modular polynomials of a given degree is finite. but I can't figure out how can we do this in general, especially, without computers.

Seewoo Lee
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    Given $E$, if $K(j(E))/K$ is an abelian extension for some $K = \mathbb{Q}(\sqrt{-d})$ and $K(j(E))/\mathbb{Q}$ is not abelian, let $q$ its conductor, if $E$ has CM it is by $R = \mathbb{Z}+ q O_K$ and it suffices to enumerate the finitely many classes of $R$-submodules of $O_K$ and compare their $j$-invariants with $j(E)$. – reuns Mar 09 '19 at 03:13
  • I don't see what you mean with the modular polynomials. The roots of the $\varphi_N(x,x)$ are the $j$-invariants of CM curves but they are not irreducible and each CM $j$-invariant appear infinitely many times. – reuns Mar 09 '19 at 04:36
  • @reuns I see, I thought they occur only finitely many times among all factors of modular polynomials. Thank you for letting me know. – Seewoo Lee Mar 10 '19 at 00:05

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