TITLE: Max order of an isogeny class of rational elliptic curves is 8?
QUESTION [8 upvotes]: 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.

REPLY [13 votes]: M. Kenku, On the number of $\mathbf{Q}$-isomorphism classes of elliptic curves in each $\mathbf{Q}$-isogeny
class, 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.