We consider two variables $t$, $u$, and the elliptic curve $E:y^{2}=x^{3}+tx+u$ which is defined over the function field $\mathcal{K}=\mathbb{C}(t,u)$. For integer $N>1$, we define the field of definition $\mathcal{K}_{N}=\mathcal{K}(E[N])$ and the Galois group $G_{N}=\mathrm{Gal}(\mathcal{K}_{N}/\mathcal{K})$. We may consider $G_{N}$ as a subgroup of $\mathrm{GL}_{2}(\mathbb{Z}/N\mathbb{Z})$. We consturct the injective homomorphism $\rho:G_{N}\to \mathrm{GL}_{2}(\mathbb{Z}/N\mathbb{Z})$ for fixed basis $\{P,Q\}$ of $E[N]$. For $\sigma\in G_{N}$, there are $a,b,c,d\in\mathbb{Z}/N\mathbb{Z}$ such that $\sigma P=[a]P+[b]Q$ and $\sigma Q=[c]P+[d]Q$. $\rho$ maps $\sigma$ to the matrix $\begin{pmatrix} a&b\\c&d\end{pmatrix}$.
My guess is that $\rho(G_{N})=\mathrm{SL}_{2}(\mathbb{Z}/N\mathbb{Z})$. My guess is based on Corollary 7.5.3 in『First Course in Modular Forms』(Diamond, Shurman). The proposition follows.
For a variable $j$ and the elliptic curve $A:y^{2}=4x^{3}-\frac{27j}{j-1728}x-\frac{27j}{j-1728}$ defined over $\mathcal{L}=\mathbb{C}(j)$,
$\rho(\mathrm{Gal}(\frac{\mathcal{L}(A[N])}{\mathcal{L}}))=\mathrm{SL}_{2}(\mathbb{Z}/N\mathbb{Z})$.
Is there a reference for my guess $\rho(G_{N})=\mathrm{SL}_{2}(\mathbb{Z}/N\mathbb{Z})$?