Bielliptic modular curves [formula omitted

Abstract

Let N≥1 be a integer such that the modular curve X0⁎(N) has genus ≥2. We prove that X0⁎(N) is bielliptic exactly for 69 values of N. In particular, we obtain that X0⁎(N) is bielliptic over the base field for all these values of N, except X0⁎(160) that is not bielliptic over Q but it does over Q(−1). Moreover, we prove that the set of all quadratic points over Q for the modular curve X0⁎(N) is infinite exactly for 100 values of N.