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.



Max order of an isogeny class of rational elliptic curves is 8?
M. Kenku, [On the number of $\mathbf{Q}$-isomorphism classes of elliptic curves in each $\mathbf{Q}$-isogeny
class](https://core.ac.uk/download/pdf/82233672.pdf), J. Number Theory **15**, 199 (1982):  

*It is shown that there are at most eight $\mathbf{Q}$-isomorphism classes of elliptic curves in each $\mathbf{Q}$-isogeny class.*


