I was trying to go through an exercise ($7.9.3$) in the book "A first course in modular forms" (Diamond, Shurman). We are trying to proof that the hecke operators $T_p$ are defined over $\mathbb{Q}$, I give now some notation: \begin{equation} \begin{split} \Gamma(N)=\{\gamma\in\text{SL}_2(\mathbb{Z}): \gamma\equiv\left[\begin{matrix}1 &0 \\ 0& 1\end{matrix}\right] \text{mod}(N)\}\\ \Gamma_0(N)=\{\gamma\in\text{SL}_2(\mathbb{Z}): \gamma\equiv\left[\begin{matrix}a &b \\ 0& c\end{matrix}\right] \text{mod}(N)\}\\ \Gamma_1(N)=\{\gamma\in\text{SL}_2(\mathbb{Z}): \gamma\equiv\left[\begin{matrix}1 &c \\ 0& 1\end{matrix}\right] \text{mod}(N)\}\\ \Gamma_{1,0}(N,p)=\Gamma_1(N)\cap\Gamma_0(p) \end{split} \end{equation} We denote respectively with $X(N)$, $X_0(N)$, $X_1(N)$ and $X_{1,0}(N,p)$ the associated modular curves. We know that all these curves are defined over $\mathbb{Q}$, we called them $X_{\dots}(\dots)_{alg}$. The point (d) is where I'm stuck, I'll cite the text
Let $\tau'=p\tau$ and let $j'=j(\tau')$, both in $\mathbb{Q}(X_{1,0}(N,p)_{alg})$ and thus is $\mathbb{Q}(j,E_j[Np])$. Show that there is a morphism $\phi\colon E_j\rightarrow E_{j'}$ with kernel $<Q_{\tau,N}>$, taking $Q_{\tau,Np}$ to $Q_{\tau',N}$ and defined over some Galois extension $\mathbb{L}$ of $\mathbb{Q}(j,E_j[Np])$. Here the second subscript of each $Q$ denots its order.
$E_j$ was defined as a curve over $\mathbb{C}(j)$ by the equation $y^2=4x^3-\frac{27j}{j-1728}x-\frac{27j}{j-1728}$ and \begin{equation} Q_{\tau,N}=\bigg(\frac{g_2(\tau)}{g_3{\tau}}\wp_\tau(1/N),\bigg(\frac{g_2(\tau)}{g_3{\tau}}\bigg)^{3/2}\wp_\tau'(1/N)\bigg) \end{equation}
I thank in advance for any help.