Cyclotomic fields and singular moduli

Let $\mu$ be the roots of unity and $S$ be the image under the modular $j$-function of all imaginary quadratic $\tau$. Then what is $\mathbb{Q}(\mu)\cap\mathbb{Q}(S)$?

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This is a subcase of: mathoverflow.net/questions/15781/… –  Dror Speiser Oct 3 '11 at 16:43
I think it isn't a subcase of that question, unless OP only wants elliptic curves with CM by the maximal order. Also, OP, do you want elliptic curves over Q, or over the algebraic closure of Q? –  Hunter Brooks Oct 3 '11 at 16:50
I think it is a subcase, but I'm not certain: I think the answer is, kind of written in the other thread, that the $j$-invariant is in $\mathbb{Q}^\text{cyc}$ if and only if the class group of the order is an elementary abelian 2-group. And I think he means over the closure, which is the same as over $\mathbb{C}$. –  Dror Speiser Oct 3 '11 at 16:59
I've edited the question to hopefully clear things up –  Adam Harris Oct 3 '11 at 17:28
It's not clear to me why this is a subcase. Could anyone please expand a little? –  Adam Harris Oct 4 '11 at 23:27