Computing quadratic points on modular curves 𝑋₀(𝑁)

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 ) X_0(N) of genus up to 8 8 , and genus up to 10 10 with N N prime, for which they were previously unknown. The values of N N we consider are contained in the set L = { 58 , 68 , 74 , 76 , 80 , 85 , 97 , 98 , 100 , 103 , 107 , 109 , 113 , 121 , 127 } . \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 ) X_0(N) for N ∈ L N\in \mathcal {L} are complex multiplication (CM) points, except for one pair of Galois conjugate points on X 0 ( 103 ) X_0(103) defined over Q ( 2885 ) \mathbb {Q}(\sqrt {2885}) . We also compute the j j -invariants of the elliptic curves parametrised by these points, and for the CM points determine their geometric endomorphism rings.