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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.

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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.

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  • $\begingroup$ Thank you, Carlo! $\endgroup$
    – ABarrios
    Commented Aug 10, 2019 at 14:21

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