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 X₀(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 L = {58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127}. We obtain that all the non-cuspidal quadratic points on X₀(N) for N ∈ L are complex multiplication (CM) points, except for one pair of Galois conjugate points on X₀(103) defined over Q(√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.