We complete the computation of all $$\mathbb {Q}$$ Q -rational points on all the 64 maximal Atkin-Lehner quotients $$X_0(N)^*$$ X 0 ( N ) ∗ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty–Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method (from a forthcoming preprint of the fourth-named author) combined with the Mordell-Weil sieve. Additionally, for square-free levels N, we classify all $$\mathbb {Q}$$ Q -rational points as cusps, CM points (including their CM field and j-invariants) and exceptional ones. We further indicate how to use this to compute the $$\mathbb {Q}$$ Q -rational points on all of their modular coverings.

Building on the work of Mazur on prime degree isogenies, Kenku determined in 1981 all the possible cyclic isogeny degrees of elliptic curves over Q. Although more than 40 years have passed, the possible isogeny degrees have not been determined over a single number field since.In this paper we determine such a classification of all possible cyclic isogeny degrees for the quadratic field Q( √ 213), assuming the Generalised Riemann Hypothesis. Along the way we determine all of the finitely many quadratic points on the modular curves X 0 (125) and X 0 (169).

Building on Mazur’s 1978 work on prime degree isogenies, Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over Q \mathbb {Q} . Although more than 40 years have passed, the determination of cyclic isogenies of elliptic curves over a single other number field has hitherto not been realised. In this paper we develop a procedure to assist in establishing such a determination for a given quadratic field. Executing this procedure on all quadratic fields Q ( d ) \mathbb {Q}(\sqrt {d}) with | d | > 10 4 |d| > 10^4 we obtain, conditional on the Generalised Riemann Hypothesis, the determination of cyclic isogenies of elliptic curves over 19 19 quadratic fields, including Q ( 213 ) \mathbb {Q}(\sqrt {213}) and Q ( − 2289 ) \mathbb {Q}(\sqrt {-2289}) . To make this procedure work, we determine all of the finitely many quadratic points on the modular curves X 0 ( 125 ) X_0(125) and X 0 ( 169 ) X_0(169) , which may be of independent interest.

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