Let $E:y^2=x^3+x$ be an elliptic curve. There is natural injection $End(E)→End( \tilde{E})$, which sends $\phi$ to it's reduction by $mod5$ (isogeny made by reduction of coefficients). But $(x,y)→(-x,iy)$ goes to $(x,y)→(-x,2y)$, $(x,y)→(-x,3y)$, does't this show the correspondence $End(E)→End( \tilde{E})$ is not well defined ?
Where am I missing ?