We complete the computation of all $$\mathbb {Q}$$Q -rational points on all the 64 maximal Atkin-Lehner quotients $$X_0(N)^*$$X0(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 levelsN, we classify all $$\mathbb {Q}$$Q -rational points as cusps, CM points (including their CM field andj-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.

We use the method of quadratic Chabauty on the quotients $X_0^+(N)$ of modular curves $X_0(N)$ by their Fricke involutions to provably compute all the rational points of these curves for prime levels $N$ of genus 4, 5, and 6. We find that the only such cu

In this paper we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curves $X_0(N)$ of genus up to $8$ , and genus up to $10$ with $N$ prime, for which they were previously unknown. The values of $N$ we consider are contained in the set $\begin{equation*} \mathcal {L}=\{58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127 \}. \end{equation*}$ We obtain that all the non-cuspidal quadratic points on $X_0(N)$ for $N\in \mathcal {L}$ are complex multiplication (CM) points, except for one pair of Galois conjugate points on $X_0(103)$ defined over $\mathbb {Q}(\sqrt {2885})$ . We also compute the $j$ -invariants of the elliptic curves parametrised by these points, and for the CM points determine their geometric endomorphism rings.