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Let $E$ be an elliptic curve of conductor $N\cdot p^2$ over $\mathbb{Q}$, defined by the equation $$y^2=x^3+p^2b\cdot x + p^3\cdot c$$ and parametrized by a map $$X_{0}(N\cdot {p}^{2})\rightarrow E$$ and let $E_{\sqrt(p)}$ be the curve $$y^2=x^3+b\cdot x+c$$ parametrized by a map $$X_{0}(N)\rightarrow E_{\sqrt(p)}$$ (Here, $p$ is a prime greater than or equal to $5$, and we assume that $E_{\sqrt(p)}$ reduces well at $p$.)

Both become isomorphic over $\mathbb{Q}[\sqrt{p}]$. Is there a lift of that isomorphism to a morphism: $$X_{0}(N\cdot {p}^{2})\rightarrow X_{0}(N)$$ or $$X_{0}(N)\rightarrow X_{0}(N\cdot {p}^{2})$$ and if so, can that morphism be described explicitly?

If it is not possible in general, can we classify the cases where such a lift exists?

Let $E$ be an elliptic curve of conductor $N\cdot p^2$ over $\mathbb{Q}$, defined by the equation $$y^2=x^3+p^2b\cdot x + p^3\cdot c$$ and parametrized by a map $$X_{0}(N\cdot {p}^{2})\rightarrow E$$ and let $E_{\sqrt(p)}$ be the curve $$y^2=x^3+b\cdot x+c$$ parametrized by a map $$X_{0}(N)\rightarrow E_{\sqrt(p)}$$ (Here, $p$ is a prime greater than or equal to $5$, and we assume that $E_{\sqrt(p)}$ reduces well at $p$.)

Both become isomorphic over $\mathbb{Q}[\sqrt{p}]$. Is there a lift of that isomorphism to a morphism: $$X_{0}(N\cdot {p}^{2})\rightarrow X_{0}(N)$$ or $$X_{0}(N)\rightarrow X_{0}(N\cdot {p}^{2})$$ and if so, can that morphism be described explicitly?

If it is not possible in general, can we classify the cases where such a lift exists?

(EDIT: If this question is stupid, feel free to tell me and to tell me why.)

Let $E$ be an elliptic curve of conductor $N\cdot p^2$ over $\mathbb{Q}$, defined by the equation $$y^2=x^3+p^2b\cdot x + p^3\cdot c$$ and parametrized by a map $$X_{0}(N\cdot {p}^{2})\rightarrow E$$ and let $E_{\sqrt(p)}$ be the curve $$y^2=x^3+b\cdot x+c$$ parametrized by a map $$X_{0}(N)\rightarrow E_{\sqrt(p)}$$ (Here, $p$ is a prime greater than or equal to $5$, and we assume that $E_{\sqrt(p)}$ reduces well at $p$.)

Both become isomorphic over $\mathbb{Q}[\sqrt{p}]$. Is there a lift of that isomorphism to a morphism: $$X_{0}(N\cdot {p}^{2})\rightarrow X_{0}(N)$$ or $$X_{0}(N)\rightarrow X_{0}(N\cdot {p}^{2})$$ and if so, can that morphism be described explicitly?

Let $E$ be an elliptic curve of conductor $N\cdot p^2$ over $\mathbb{Q}$, defined by the equation $$y^2=x^3+p^2b\cdot x + p^3\cdot c$$ and parametrized by a map $$X_{0}(N\cdot {p}^{2})\rightarrow E$$ and let $E_{\sqrt(p)}$ be the curve $$y^2=x^3+b\cdot x+c$$ parametrized by a map $$X_{0}(N)\rightarrow E_{\sqrt(p)}$$ (Here, $p$ is a prime greater than or equal to $5$, and we assume that $E_{\sqrt(p)}$ reduces well at $p$.)

Both become isomorphic over $\mathbb{Q}[\sqrt{p}]$. Is there a lift of that isomorphism to a morphism: $$X_{0}(N\cdot {p}^{2})\rightarrow X_{0}(N)$$ or $$X_{0}(N)\rightarrow X_{0}(N\cdot {p}^{2})$$ and if so, can that morphism be described explicitly?

If it is not possible in general, can we classify the cases where such a lift exists?