Speaker
Description
We determine all possible degrees of cyclic isogenies of elliptic curves with rational $j$-invariant defined over degree $d$ extensions of $\mathbb Q$ for $d=3,5,7$.
The same question for $d=1$ has been answered by Mazur and Kenku in 1978-1982, and Vukorepa answered the question for $d=2$. All possible prime degrees of isogenies were previously found by Najman.
We use well known results about images of Galois representations of elliptic curves, as the images of those representations are closely related to the existence of isogenies. The possible images of $p$-adic representations have mostly been determined for primes of our interest, and we use these results to reduce to a finite number of cases. The remaining cases are solved by finding all rational points on some modular curves, which is done with the assistance of the computer algebra system Magma.
