I am looking for a reference for the proof of the following question following Theorem 5 in Mazur's *Rational Isogenies of Prime Degree*.

**Theorem 5** There is a constant $C$ such that every elliptic curve $E_{/\mathbb{Q}}$ is isogenous (over $\mathbb{Q}$) to at most $C$ (mutually nonisomorphic) elliptic curves.

"Can one take $C=8$?"

Has this question been settled? And if so, what is a reference to the proof of the result.