  The main goal of this paper is to determine for which prime numbers $r\geq 3$
can an elliptic curve~$E$ defined over $\mathbb Q$ have an $r$-isogeny over
$\mathbb Q(\zeta_r)$. We study this question under various assumptions on the
2-torsion of $E$. Apart from being a natural question itself, the mod~$r$
representations attached to such $E$ arise in the Darmon program for the
generalized Fermat equation of signature $(r,r,p)$, playing a key role in the
proof of modularity of certain Frey varieties in the recent work of Billerey,
Chen, Dieulefait and Freitas.
