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In the proof of part c), I cannot make sense of the sentence "Then another way of saying that $\phi:E_1\to E_2$ is an isogeny is to note that $\phi(x_1,y_1)\in E_2(K(x_1,y_1))$."

(In this situation, $\phi$ should be an isogeny between elliptic curves $E_1$ and $E_2$ originally defined over $K$, but for the sake of this sentence they are considered over $K(E_1)=K(x_1,y_1)$, where $x_1$ and $y_1$ are Weierstrass-coordinates for $E_1$.)

I don't see how the two statements are related, and moreover it seems to me that the latte statement would be true, if $\phi$ was just a morphism.

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This has been noted before. See "Why is this a characterization of isogenies of elliptic curves? (From Silverman)". The statement $\phi(x_1,y_1)\in E_2(K(x_1,y_1))$ indeed just says that $\phi$ is a morphism landing on $E_2$ (which is true since $\phi$ is an isogeny by assumption).

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