  We study the finiteness of low degree points on certain modular curves and
their Atkin--Lehner quotients, and, as an application, prove the modularity of
elliptic curves over all but finitely many totally real fields of degree $5$.
On the way, we prove a criterion for the finiteness of rational points of
degree $5$ on a curve of large genus over a number field using the results of
Abramovich--Harris and Faltings on subvarieties of Jacobians.
