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  1. Given $K$ a number field and $E/K$ an elliptic curve, is there an algorithm which gives all the elliptic curves $F/K$ isogenous to $E$ (up to isomorphism)?

  2. Or is there a bound on how many $F/K$ are isogenous to $E/K$?

  3. If $E/K$ and $F/K$ are isogenous is the isogeny necessarily defined over $K$?

For a fixed degree, I know we can use modular polynomials to find $F/K$. However is there a bound on the degree? I know that when $K=\mathbb{Q}$, this follows from Mazur's theorem.

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  • $\begingroup$ My answer to this question contains links to algorithms even over number fields. That should help for (1). $\endgroup$
    – Chris Wuthrich
    Commented Oct 20 at 15:31
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    $\begingroup$ (3) No, because of quadratic twists this isn't true even for isomorphic curves (degree 1 isogenies). $\endgroup$
    – Chris Wuthrich
    Commented Oct 20 at 15:32
  • $\begingroup$ Thank you. I have a follow-up naive question. In this algorithm, they show how to find primes $l$ such that $E/K$ has an $l$-isogeny. It seems to me that in this context the $l$-isogenies are thought to be defined over $K$, is that right? $\endgroup$
    – did
    Commented Oct 20 at 18:27
  • $\begingroup$ The algorithms look for all isogenies $\phi$ leaving $E$ defined over a fixed number field $K$, which is equivalent to finding their kernels $C = E[\phi]$ as subgroup of $E$ defined over $K$. The codomain $E/C$ is defined over $K$. $\endgroup$
    – Chris Wuthrich
    Commented Oct 22 at 11:46
  • $\begingroup$ If $\phi: E \to E'$ is an isogeny defined over $F$ between two curves defined over $K$, then $\phi$ is the composition of an isogeny of that degree defined over $K$ and an isomorphism from $E/E[\phi]$ to $E'$ defined over $F$. $\endgroup$
    – Chris Wuthrich
    Commented Oct 22 at 11:46

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