  Modular Galois representations into GL_2(F_p) with cyclotomic determinant
arise from elliptic curves for p = 2,3,5. We show (by constructing explicit
examples) that such elliptic curves cannot be chosen to have conductor as small
as possible at all primes other than p. Our proof involves finding all elliptic
curves of conductor 85779, a custom computation carried out for us by Cremona.
This leads to a counterexample to a conjecture of Lario and Rio. For p > 5, we
construct irreducible representations with cyclotomic determinant that do not
arise from any elliptic curve over Q.
