It is known that if p > 37 is a prime number and E/Q is an elliptic curve without complex multiplication, then the image of the mod p Galois representation ρE,p :

of E is either the whole of GL2(Fp), or is contained in the normaliser of a non-split Cartan subgroup of GL2(Fp). In this paper, we build on work of Zywina and show that when p > 1.4 × 10 7 , the image of ρE,p is either GL2(Fp), or is the normaliser of a non-split Cartan subgroup. We use this to show the following result, partially settling a question of Najman. For d ≥ 1, let I(d) denote the set of primes p for which there exists an elliptic curve defined over Q and without complex multiplication admitting a degree p isogeny defined over a number field of degree ≤ d. We show that, for d ≥ 1.4 × 10 7 , we have