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I have something don't understand about the CM types of CM elliptic curves. I want to determine the CM type of certain elliptic curves. Let $K=\mathbb{Q}(\sqrt{-3})$ be the CM field and $\omega=\frac{-1}{2}+\frac{\sqrt{-3}}{2}$ the third root of unity. Let $E:y^2=x^3-\omega$ and $E:y^2=x^3-\omega^2$. Both $E$ and $E'$ have the CM by $K$. Is their CM type the same? How to determine their CM type exactly?

Maybe, even more typically, what is the CM type of $E'': y^2=x^3+1$. Does the CM type depend on something?

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  • $\begingroup$ what do you mean by CM-type? when people say CM-type of an abelian variety they usually mean $(K,\Phi)$ where $K$ is a CM-field and $\Phi$ is a decomposition of embeddings of K into $\mathbb{C}$ satisfy some condition. but in the case of elliptic curve for each $K$ you have only one choice for $\Phi$ so you can ignore it. $\endgroup$
    – ali
    Commented May 18, 2022 at 13:38
  • $\begingroup$ Maybe not. $K$ has two embeddings into $\mathbb{C}$, conjugate to each other. So it has two CM-type. CM-type here should mean which embedding $\iota$ we use to get the isogeny from $E$ to $\mathbb{C}/\iota(O_K)$. $\endgroup$
    – yhb
    Commented May 18, 2022 at 23:57

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