= Solution
The weights of $E_4^3$ and $\Delta$ agree, so their ratio is invariant under the <modular group>. Nonvanishing of the <modular discriminant> on the half-plane makes the ratio holomorphic there. At an elliptic point, an invariant <holomorphic function> has a power series in the quotient coordinate, so the ratio descends holomorphically. At the unique <modular cusp> its expansion is
$$
j=q^{-1}+O(1).
$$
Thus it is a <meromorphic function> on the compact <Riemann surface> with precisely one <pole>, of order one. The degree of a nonconstant meromorphic map to the sphere equals its total <pole> order. It therefore has degree one, hence is a <biholomorphism>:
$$
\boxed{j:X(\Gamma(1))\overset{\sim}{\longrightarrow}\mathbb P^1_{\mathbb C}.}
$$
This is the <single-pole criterion for a spherical coordinate>.
The <matrix> $\begin{pmatrix}0&-1\\1&1\end{pmatrix}$ fixes $\omega$. Weight-four transformation gives $E_4(\omega)=(\omega+1)^4E_4(\omega)$, and $(\omega+1)^4\ne1$. Hence $E_4(\omega)=0$ and \b[$j(\omega)=0$]. Since $j$ is a degree-one coordinate, this is a simple zero on the quotient surface. Pulled back to the half-plane it has order three, by the <analytic ramification index> in part (a).
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