We give explicit uniform bounds for several quantities relevant to the study of Galois representations attached to elliptic curves E/ Q. We consider in particular the subgroup of scalars in the image of Galois, the first Galois cohomology group with values in the torsion of E, and the Kummer extensions generated by points of infinite order in E( Q).
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Lombardo et al. (2022) studied this question.
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