  We study the splitting of primes in number fields generated by points on
modular curves. Momose was the first to notice that quadratic points on
$X_1(n)$ generate quadratic fields over which certain primes split in a
particular way and his results were later expanded upon by Krumm. We prove
results about the splitting behaviour of primes in quadratic fields generated
by points on the modular curves $X_0(n)$ which are hyperelliptic (except for
$n=37$) and in cubic fields generated by points on $X_1(2, 14)$.
