Constructing elliptic curves defined over $\mathbb{Q}$ with fixed complex multiplication

Question

I have the following problem: we know that for a field $\kappa$ of characteristic $0$ usually an elliptic curve $E$ defined over $\kappa$ is such that $End(E)\cong \mathbb{Z}$. This means that one cannot hope to find an elliptic curve with complex multiplication choosing it "randomly".

Suppose that i want to produce and elliptic curve over $\mathbb{Q}$ whose $End(E)$ is and order in $\mathbb{Q}(\sqrt{-D})$: what are the methods currently known to do it? Is it possible to write explicitly the isogeny corresponding to $\sqrt{-D}$? (If it is in $End(E)$ and $D$ is not so large).

Thank you for your time.

Answer 1

There won''t be an $E/\mathbb Q$ with CM by an order $R$ in $\mathbb Q(\sqrt{-D})$ unless $R$ has class number 1, and there are only finitely many such orders. There is a complete list of the corresponding elliptic curves (probably already since the 19th century, modulo the problem that they didn't know the non-existence of a 10th field of class number 1). The isogenies won't be defined over $\mathbb Q$, but certainly one can write them down, fairly easily for the smaller values of $D$. Although I don't think I've seen them written out for the larger values such as $\frac12(1+\sqrt{-163})$ isogeny written down explicitly.

Equations for elliptic curves over $\mathbb Q$ that have CM are listed in [1, Appendix A, Section 2]. If you want to see how one can explicitly find curves and equations for the isogenies $\sqrt{-2}$ and $\frac12(1+\sqrt{-7})$, see [1, Chapter II, Section 2], and in particular Proposition II.2.3.1.

[1] Advanced Topics in the Arithmetic of Elliptic Curves, Springer, New York, 1994.