Title: an effective open image theorem for elliptic curves over the rationals without CM Abstract: A crucial hypothesis to apply Kolyvagin's theorem is the surjectivity of the mod-p Galois representation of E/Q without CM. We prove an explicit bound p_0(N) only depending on the conductor N of E such that the mod-p Galois representation is surjective if p > p_0(N). The proof is based on an argument due to Serre, which uses Mazur's classification of rational isogenies of prime degree, the density of supersingular primes for E/Q and the modularity of elliptic curves over the rationals.