Summary: It is known that if \(p>37\) is a prime number and \(E/\mathbb{Q}\) is an elliptic curve without complex multiplication, then the image of the mod \(p\) Galois representation \[ \bar{\rho}_{E,p}:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \mathrm{GL}(E[p]) \] of \(E\) is either the whole of \(\mathrm{GL}(E[p])\), or is contained in the normaliser of a nonsplit Cartan subgroup of \(\mathrm{GL}(E[p])\). In this paper, we show that when \(p>1.4\times 10^7\), the image of \(\bar{\rho}_{E,p}\) is either \(\mathrm{GL}(E[p])\), or the full normaliser of a nonsplit Cartan subgroup. We use this to show the following result, partially settling a question of Najman. For \(d\geq 1\), let \(I(d)\) denote the set of primes \(p\) for which there exists an elliptic curve defined over \(\mathbb{Q}\) and without complex multiplication admitting a degree \(p\) isogeny defined over a number field of degree \(\leq d\). We show that, for \(d\geq 1.4\times 10^7\), we have \[ I(d)=\{p\text{ prime}:p\leq d-1\}. \]