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According the the introduction to Mazur's Rational Isogenies of Prime Degree the following question was open in 1978:

Let $N$ be one of the integers 39, 65, 91, 125, or 169. Does the modular curve $X_0(N)$ possess noncuspidal rational points?

It seems likely that this should have been resolved in the past 32 years. Does anyone know of a reference for this?

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2 Answers

up vote 6 down vote accepted

Let me make David Brown's answer more explicit.

Theorem (Kenku, 1979): $X_0(39)(\mathbb{Q})$ consists entirely of the ($4$) cuspidal points.

Theorem (Kenku, 1980): $X_0(65)(\mathbb{Q})$ and $X_0(91)(\mathbb{Q})$ each consist entirely of the ($4$) cuspidal points.

Theorem (Kenku, 1980, see also Kenku, 1980): $X_0(169)(\mathbb{Q})$ consists entirely of $2$ rational cuspidal points.

Theorem (Kenku, 1981): $X_0(125)(\mathbb{Q})$ consists entirely of $2$ rational cuspidal points.

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These are all due to Kenku; see mathscinet.

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