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Let $F_{N,\mathbb C}$ be the field of modular functions of level $N$. We have $$\operatorname{Gal}(F_{N,\mathbb C}/\mathbb C(j))\cong \operatorname{SL}_2(\mathbb Z/N\mathbb Z)/\{\pm 1\}.$$

Let $f\in F_{N,\mathbb C}$. To be able to do something with $f$, I must assume that $\mathbb C(j)\subset\mathbb C(f)$.

Is there a modular function of some level, such that this assumption is not true? Or is the $j$-invariant always rationally expressible through every modular function?

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    $\begingroup$ The modular curve $X(N)$ is the compactification of $\mathbb{H}/\Gamma (N)$; it always maps to the compactification of $\mathbb{H}/\operatorname{PSL}_2 (\mathbb{Z})$, which is isomorphic to $\mathbb{P}^1$, the isomorphism being given by the $j$-function. Thus the field $F_N=\mathbb{C}(X(N))$ is a finite extension of $\mathbb{C}(j)$. $\endgroup$ – abx Apr 27 '19 at 19:24
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    $\begingroup$ Take $f=j^2$. Then $\mathbb{C}(f) = \mathbb{C}(j^2) \not\ni j$. (Geometrically, take two points $\tau_1$ and $\tau_2$ with $j(\tau_1) = -j(\tau_2) \neq 0$. Any rational function of $j^2$ will take the same value at $\tau_1$ and $\tau_2$.) $\endgroup$ – David E Speyer Apr 27 '19 at 20:01
  • $\begingroup$ We can see already in the case $N=1$ that it doesn't hold. $\endgroup$ – François Brunault Apr 28 '19 at 9:13

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