I have some questions about isogenies of elliptic curves in two settings:

**1. Elliptic curves defined over the rationals.**

1.1. Given two elliptic curves $E/\mathbb{Q}$ and $E'/\mathbb{Q}$ we can decide if they are $\mathbb{Q}$-isogenous. One way to do it is to use Mazur's theorem to bound the degree of possible isogenies for $E$. Then we can construct all isogenies up to that degree (can this be done by Velu's formulæ?) and the corresponding isogenous curves (up to isomorphism). Finally, we can check if $E'$ is isomorphic to one of these (finitely many) curves. I believe Magma can do this.

1.2. It seems to me that the above procedure outputs the isogeny between $E/\mathbb{Q}$ and $E'/\mathbb{Q}$ if they are isogenous.

**2. Elliptic Curves over a finite field $k$**

2.1. Deciding if two elliptic curves are isogenous is easy due to Tate's isogeny theorem which says that elliptic curves over finite fields are isogenous iff the order of their set of $k$-points is the same.

2.2. Is there any way to tell the degree of this isogeny? Is there any way to list all of the isogenies of an elliptic curve over a finite field? I know that computing the isogeny itself is very difficult as many cryptosystems depend precisely on the hardness of this problem. However, I would be interested in what is the best that we can do, what are the techniques and what are the difficulties (in comparison to the rational case).