Images of \(\text{mod }p\) Galois representations associated to elliptic curves (Q2764789)

Let \(E\) be an elliptic curve over a number field \(K\). The authors study the image of the representation of the absolute Galois group of \(K\) acting on \(p\)-torsion points of \(E(\overline{K}),\) where \(p\) is a prime number. If \(E\) does not admit complex multiplication then it is well known that this image is GL\((2,{\mathbb F}_p)\) for all but finitely many primes \(p\). The authors first derive certain restrictions on the image when there exists an isogeny of \(E\) of degree \(p\). Then the case \(p\in\{2,3\}\) is treated separately. The paper finally culminates in a list describing the images of all Galois representations for all of the 722 elliptic curves over the rationals of conductor no more than 200 without complex multiplication.