Remarks on $X_0(N)$ (awaiting review)

Hyperelliptic curves: There are only 19 hyperelliptic curves in this family [10.24033/bsmf.1789, MR:0364259], [10.5802/aif.1273, MR:1150566].

Bielliptic curves: There are only 41 bielliptic curves in this family [10.1006/jnth.1998.2343, MR:1688168], [arXiv:1207.2009, MR:3086199]. Among them, there are 30 values for which they are non-hyperelliptic.

Rational points: There are only 26 curves in this family that have non-cuspidal rational points [10.1007/BF01390348, MR:0482230], [10.1112/jlms/s2-23.3.415, MR:0616546].