Let $E$ be an elliptic curve of conductor $N\cdot p^3$ over $\mathbb{Q}$, defined by the equation
$$y^2=x^3+p^2b\cdot x + p^3\cdot c$$
and parametrized by a map
$$X_{0}(p\cdot N)\rightarrow E$$
and let $E_{\sqrt(p)}$ be the curve
$$y^2=x^3+b+c$$
parametrized by a map
$$X_{0}(N)\rightarrow E_{\sqrt(p)}$$
Both become isomorphic over $\mathbb{Q}[\sqrt{p}]$. Can a lifting of this isomorphism to a morphism: $X_{0}(N\cdot {p}^{3})\rightarrow X_{0}(N)$ be described explicitly?