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

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Cyclotomic fields and singular moduli

Let $\mu$ be the roots of unity and $S$ be the set of $j$-invariants of elliptic curves with complex multiplication. Then what is $\mathbb{Q}(\mu)\cap\mathbb{Q}(S)$?