It has been proved by Bourdon, Ejder, Liu, Odumodu, and Viray that, assuming Serre's uniformity conjecture, there are finitely many non-CM j_0 \in Q such that there is an elliptic curve E with j(E)=j_0 which corresponds to an isolated point on X_1(N). If one considers the same question for general modular curves X_H for H