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

of $E$ is either the whole of $GL\left(E\left[p\right]\right)$, or is contained in the normaliser of a nonsplit Cartan subgroup of $GL\left(E\left[p\right]\right)$. In this paper, we show that when $p>1.4\times 10^7$, the image of ${\bar{\rho }}_{E,p}$ is either $GL\left(E\left[p\right]\right)$, or the full normaliser of a nonsplit Cartan subgroup. We use this to show the following result, partially settling a question of Najman. For $d\ge 1$, let $I\left(d\right)$ 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

据了解，如果 $磷>37$ 是一个质数并且 $乙∕\mathbb{Q}$ 是一条没有复杂乘法的椭圆曲线，那么mod的图像 $磷$ 伽罗瓦表示

的 $乙$ 要么是全部 $GL\left(乙\left[磷\right]\right)$，或包含在非分裂 Cartan 子群的归一化器中$GL\left(乙\left[磷\right]\right)$. 在本文中，我们表明当$磷>1.4\times 10^7$，图像 ${\bar{\rho }}_{乙,磷}$ 或者是 $GL\left(乙\left[磷\right]\right)$，或非分裂 Cartan 子群的完全归一化器。我们用它来展示以下结果，部分解决了 Najman 的问题。为了$d\ge 1$， 让 $\mathrm{一世}\left(d\right)$ 表示素数集 $磷$ 存在定义在的椭圆曲线 $\mathbb{Q}$ 并且没有复杂的乘法承认学位 $磷$ 在一定数量的度域上定义的同基因性 $\le d$. 我们证明，对于$d\ge 1.4\times 10^7$， 我们有