In Lang's book Elliptic Functions, he shows how to generate the ray class field with conductor $N$ of an imaginary quadratic number field $k$ using the $j$-invariant of an elliptic curve $A/\mathbb{C}$ with $\mathrm{End}(A)\cong\frak{o}_\text{$k$}$ and the values of the Weber function at $N$-torsion points of $A$. Namely, Theorem 2 of Chapter 10 reads as follows:
Let $A$ be an elliptic curve whose ring of endomorphisms is the ring of algebraic integers $\frak{o}_\text{$k$}$ in an imaginary quadratic number field $k$, and $A$ is defined over $k(j_A)$. Let $h$ be the Weber function on $A$, giving the quotient of $A$ by its group of automorphisms. Then $k(j_A, h(A_N))$ is the ray class field of $k$ with conductor $N$.
However, his proof starts out by saying "[l]et $K$ be the smallest Galois extension of $k$ containing $j_A=j(\frak{a})$ and all coordinates $h(A_N)$" and then he goes on to prove that $K$ is the ray class field of $k$ with conductor $N$. After that, he concludes that "[t]his proves Theorem 2", without ever mentioning the fact that we have not yet proved $K=k(j_A, h(A_N))$, i.e. we do not yet know that $k(j_A, h(A_N))$ is Galois. Is there an obvious reason I am missing here?
Notice that he does a similar thing when he proves that the Hilbert class field of $k$ is $k(j(\frak{a}))$ with $\frak{a}$ some fractional ideal of $k$: He starts by defining $K$ as the smallest Galois extension of $k$ containing all $j(\frak{a}_\text{$i$})$, where the $\frak{a}_\text{$i$}$ are a set of representatives for the ideal class group, proves that $K$ is the Hilbert class field of $k$ and then decides to be done. However, here I can conclude the argument: It was shown in the course of the proof that all $j(\frak{a}_\text{$i$})$ are conjugate, hence $j(\frak{a})$ has at least degree $h_k$ over $k$ and since this is also the degree of the Hilbert class field of $k$ over $k$, the Hilbert class field must already be $k(j(\frak{a}))$. Maybe something similar is possible for the ray class field?
EDIT: I briefly thought that one could argue via division polynomials, but the trouble is that they are only defined over $k(j_A)$ and not over $k$. Namely, if $j_A\neq 0, 1728$ then the Weber function $h$ of an $N$-torsion point $P$ is just the $x$-coordinate of $P$ and these $x$-coordinates are precisely the roots of some polynomial $\psi$ obtained from the $N$-th division polynomial, i.e. $k(j_A, h(A_N))$ is the splitting field of $\psi$. However, as $\psi$ is only defined over $k(j_A)$ and not necessarily over $k$, this only lets us conclude that $k(j_A, h(A_N))$ is Galois over $k(j_A)$ and, sadly, being Galois is not transitive ...