Residual Galois representations of elliptic curves with image contained in the normaliser of a non-split Cartan

It is known that if $p>37$ is a prime number and $E\mathrm{/}\mathbb{Q}$ is an elliptic curve without complex multiplication, then the image of the mod $p$ Galois representation $${\bar{\rho }}_{E,p}:\mathrm{Gal}(\bar{\mathbb{Q}}\mathrm{/}\mathbb{Q})\to \mathrm{GL}(E[p])$$ of $E$ is either the whole of $\mathrm{GL}(E[p])$, or is \emph{contained} in the normaliser of a non-split 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 \emph{full} normaliser of a non-split Cartan subgroup. We use this to show the following result, partially settling a question of Najman. For $d\ge 1$, 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 $\le d$. We show that, for $d\ge 1.4\times {10}^7$, we have $$I(d)=\{p\text{ prime}:p\le d-1\}\mathrm{.}$$