Let $ell$ be an odd prime and $d$ a positive integer. We determine whenthere exists a degree-$d$ number field $K$ and an elliptic curve $E/K$ with$j(E)inmathbb{Q}setminus{0,1728}$ for which $E(K)_{mathrm{tors}}$contains a point of order $ell$. We also determine when there exists such apair $(K,E)$ for which the image of the associated mod-$ell$ Galoisrepresentation is contained in a Cartan subgroup or its normalizer,conditionally on a conjecture of Sutherland. We do the same under the strongerassumption that $E$ is defined over $mathbb{Q}$.
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机译：假设$ ell $是一个奇数质数，$ d $是一个正整数。我们确定何时存在一个度数-$ d $数字字段$ K $和一个椭圆曲线$ E / K $，其中$ j（E） in mathbb {Q} setminus {0,1728 } $ E（K）_ { mathrm {tors}} $包含顺序点$ ell $。我们还确定是否存在这样的一对$（K，E）$，其相关mod-$ ell $ Galoisrepresentation的图像包含在Cartan子组或其规范化器中，有条件地取决于Sutherland的猜想。在$ E $定义在$ mathbb {Q} $上的更强假设下，我们执行相同的操作。
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