Isomorphic elliptic curves (abelian varieties) over $\mathbb{C}$ and $\overline{\mathbb{Q}}$

Question

For two elliptic curves $E_1, E_2$ defined over $\mathbb{Q}$, we assume $E_1$ and $E_2$ are isomorphic over $\mathbb{C}$, then how to prove they are isomorphic over $\overline{\mathbb{Q}}$?

Also, can we get the same conclusion if we replace elliptic curves by abelian varieties? Here, an abelian variety by definition is a complete connected group variety.

Thanks very much!

Answer 1

Let the curves be $$E_i:\qquad y^2=x^3+a_ix+b_i$$ for $i=1$, $2$. If they are isomorphic over $\Bbb C$ then there is a non-zero complex number $t$ with $a_2=t^4a_1$ and $b_2=t^6b_1$. If $a_1\ne0$ then $t^4$ is rational and $t\in\overline{\Bbb Q}$. Likewise if $b_1\ne0$. But if $a_1=b_1=0$ then $E_1$ wasn't an elliptic curve.