  Let K be a totally real Galois number field and let A be a set of elliptic
curves over K. We give sufficient conditions for the existence of a finite
computable set of rational primes P such that for p not in P and E in A, the
representation on E[p] is irreducible. Our sufficient conditions are often
satisfied for Frey elliptic curves associated to solutions of Diophantine
equations; in that context, the irreducibility of the mod p representation is a
hypothesis needed for applying level-lowering theorems. We illustrate our
approach by improving on an existing result for Fermat-type equations of
signature (13, 13, p).
