Reducible mod 105 representations and modularity of elliptic curves

Let F be a totally real number field which is unramified at 3 , 5 , and 7. We give explicit conditions on a finite place v 0 of F such that any elliptic curve over F with potential good reduction at v 0 is modular. The main point is to prove that there exist no elliptic curves with reducible mod 5, and 7 representations (or mod 3, 5, and 7 representations) under our assumptions.