Abstract
We complete the computation of all mathbb {Q}-rational points on all the 64 maximal Atkin-Lehner quotients 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}-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}-rational points on all of their modular coverings.
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