For a positive integer $N$, we define $$\Gamma(N)=\big \{ \begin{bmatrix} a & b \newline c & d\end{bmatrix}\in \operatorname{SL}_2(\mathbb{Z}): \begin{bmatrix} a & b \newline c & d\end{bmatrix}\equiv \begin{bmatrix} 1 & 0 \newline 0 & 1\end{bmatrix}(\operatorname{mod}N)\big\}$$

Then, [Diamond, Shurman] defines the set of equivalence classes $S(N)$ of enhanced elliptic curves associated to $\Gamma(N)$ to be the set of equivalence classes of all triples $(E,P,Q)$, where $E/\mathbb{C}$ is an elliptic curve, and $P,Q\in E(\mathbb{C})[N]$ such that the Weil pairing $e_N(P,Q)=e^{2\pi i/N}$ .

Defining a Weil pairing:

Let $P_1,P_2\in E(\mathbb{C})[N]$ (possibly equal). Firstly, fix $E(\mathbb{C})=\mathbb{C}/\Lambda$ for some lattice $\Lambda=\mathbb{Z}\omega_1+\mathbb{Z}\omega_2\subset \mathbb{C}$. Now, there is a matrix $\gamma\in M_{2\times 2}(\mathbb{Z}/N\mathbb{Z})$ such that $\begin{bmatrix} P_1 \\ P_2\end{bmatrix}=\gamma \begin{bmatrix} \omega_1/N+\Lambda \\ \omega_2/N+\Lambda\end{bmatrix}$. We define $e_N(P_1,P_2)=e^{2\pi i\operatorname{det}(\gamma)/N}$.

Now, I know that there exists a smooth affine curve $X_{\widetilde{\Gamma}(N)}/\operatorname{Spec}\mathbb{Z}[1/N]$ which represents the moduli problem (assume $N$ is large enough): $$X_{\widetilde{\Gamma}(N)}(S)=\{(E,\alpha):E/S \operatorname{\ is\ an\ elliptic\ curve\ }, \alpha:\underline{(\mathbb{Z}/N\mathbb{Z})^{\oplus 2}}_S\xrightarrow{\cong}E[N]\}.$$

At the face of it, it seems like $X_{\widetilde{\Gamma}(N)}(\mathbb{C})=S(N)$. But, the problem I am having is that an isomorphism $\alpha:(\mathbb{Z}/N\mathbb{Z})^{\oplus 2}\xrightarrow{\cong}E(\mathbb{C})[N]$ gives two points $P,Q\in E(\mathbb{C})[N]$ such that $e_N(P,Q)=e^{2\pi i x/N}$ for some $x\in (\mathbb{Z}/N\mathbb{Z})^\times$, which is not the same as a point in $S(N)$

Am I missing something? Are these two the same?

Reference:

*Diamond, Fred; Shurman, Jerry*, A first course in modular forms, Graduate Texts in Mathematics 228. Berlin: Springer (ISBN 0-387-23229-X/hbk). xv, 436 p. (2005). ZBL1062.11022.