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BIS HD
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Let K$K$ be an imaginary quadratic field with Hilbert class field H$H$, and let E$E$ be an elliptic curve defined over H$H$ with complex multiplication by the ring of integers $O_K$ of K$K$. It is known that for an integral ideal $\mathfrak{m}$ of $O_K$, $K(j(E),h(E[\mathfrak{m}]))$ is the ray class field of K modulo $\mathfrak{m}$, where $h$ is the Weber function for E/H$E/H$. (This is stated, for example, on page 135 of Silverman's Advanced Topics in the Arithmetic of Elliptic Curves.)

My question is this: what if E$E$ has CM by an arbitrary order? Can any generalization of this statement be made? I've read that if E$E$ has CM by an order of conductor f$\mathfrak{f}$, then $K(j(E))$ is the ring class field of $K$ with conductor f$\mathfrak{f}$, but I'm wondering if anything more can be said.

Let K be an imaginary quadratic field with Hilbert class field H, and let E be an elliptic curve defined over H with complex multiplication by the ring of integers $O_K$ of K. It is known that for an integral ideal $\mathfrak{m}$ of $O_K$, $K(j(E),h(E[\mathfrak{m}]))$ is the ray class field of K modulo $\mathfrak{m}$, where $h$ is the Weber function for E/H. (This is stated, for example, on page 135 of Silverman's Advanced Topics in the Arithmetic of Elliptic Curves.)

My question is this: what if E has CM by an arbitrary order? Can any generalization of this statement be made? I've read that if E has CM by an order of conductor f, then $K(j(E))$ is the ring class field of $K$ with conductor f, but I'm wondering if anything more can be said.

Let $K$ be an imaginary quadratic field with Hilbert class field $H$, and let $E$ be an elliptic curve defined over $H$ with complex multiplication by the ring of integers $O_K$ of $K$. It is known that for an integral ideal $\mathfrak{m}$ of $O_K$, $K(j(E),h(E[\mathfrak{m}]))$ is the ray class field of K modulo $\mathfrak{m}$, where $h$ is the Weber function for $E/H$. (This is stated, for example, on page 135 of Silverman's Advanced Topics in the Arithmetic of Elliptic Curves.)

My question is this: what if $E$ has CM by an arbitrary order? Can any generalization of this statement be made? I've read that if $E$ has CM by an order of conductor $\mathfrak{f}$, then $K(j(E))$ is the ring class field of $K$ with conductor $\mathfrak{f}$, but I'm wondering if anything more can be said.

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abourdon
abourdon
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Elliptic Curves with CM and Class Field Theory

Let K be an imaginary quadratic field with Hilbert class field H, and let E be an elliptic curve defined over H with complex multiplication by the ring of integers $O_K$ of K. It is known that for an integral ideal $\mathfrak{m}$ of $O_K$, $K(j(E),h(E[\mathfrak{m}]))$ is the ray class field of K modulo $\mathfrak{m}$, where $h$ is the Weber function for E/H. (This is stated, for example, on page 135 of Silverman's Advanced Topics in the Arithmetic of Elliptic Curves.)

My question is this: what if E has CM by an arbitrary order? Can any generalization of this statement be made? I've read that if E has CM by an order of conductor f, then $K(j(E))$ is the ring class field of $K$ with conductor f, but I'm wondering if anything more can be said.