Dear Barinder,

Are you familiar with Fumiyuki Momose's "Isogenies of prime degrees over number fields?" If not, you may find it here on [NUMDAM][1] In it he performs an analysis of the isogeny character and finds that if $k$ is a quadratic field which is not a class number one imaginary quadratic field there are only finitely many $p$ for which $X_0(p)$ has noncuspidal rational points.


  [1]: http://www.numdam.org/numdam-bin/item?id=CM_1995__97_3_329_0