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Let K be a number field. The Gal(
/K)-action on the torsion of an elliptic curve E/K gives rise to an adelic representation ρ
E
: Gal(/K) ⇉ GL2(
). From an analysis of maximal closed subgroups of GL2() we derive useful necessary and sufficient conditions for ρ
E
to be surjective. Using these conditions, we compute an example of a number field K and an elliptic curve E/K that admits a surjective adelic Galois representation.
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