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The following question is in my homework:

How many complex elliptic curves (up to isomorphism) have complex multiplication by the ring $\mathbb{Z}[\frac{1+\sqrt{D}}{2}]$ of discriminant $D=-71$ and and of discriminant $D=-163$.

I do not even know where to start and cannot really find a good reference on complex multiplication in elliptic curves. Silverman's books seem to only provide a definition and very advanced work and I don't know how to use the information to approach this problem. There are several problems in this homework set which require complex multiplication, so some help on this topic would be much appriciated!

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Hint: any ideal $I$ of this ring $R=\mathbf{Z}\left[\frac{1+\sqrt{D}}{2}\right]$ can be viewed as a lattice sitting inside $\mathbf C$. Moreover, multiplication by elements of $R$ preserves this lattice, so it descends to the quotient $\mathbf C/I$. In other words, $\mathbf C/I$ has complex multiplication by $R$. Now, try to answer the following questions:

  1. When do two ideals $I, I'$ give isomorphic elliptic curves $\mathbf C/I, \mathbf C/I'$?
  2. Do you get all isomorphism classes of elliptic curves with complex multiplication by $R$ in this way?
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