There are two questions related to elliptic curve
(1)Let E be an algebraic elliptic curve over k where k:algebraic closed field with characteristic zero
Let K be an any extension field of k so we can view E as an elliptic curve over K
Then is it right that $E(k)^{tor}=E(K)^{tor}?$
$E(k)[N]\subset E(K)[N]$ and $|E(k)[N]|=N^2=|E(\overline K)[N]|$ implies that E(k)[N]=E(K)[N]
So we can conclude that $E(k)^{tor}=E(K)^{tor}?$ Is it right?
(2)Let $E_1, E_2$ be algebraic elliptic curves over $\overline{\mathbb{Q}}$
Assume that $E_1, E_2$ are isomorphic over $\mathbb{C}$(View $E_1, E_2$ elliptic curve over $\mathbb{C}$)
Can we show that this morphism is actually defined over $\overline{\mathbb{Q}}$?
It looks like that this statement hold for any algebraic curve $C_1, C_2$
Thank you.