Suppose I have a motive $M$ over $\mathbb{Q}$, and can compute the Euler factor of the associated $L$-function for any good prime $p$. How can I compute the Zariski closure of the image of the Galois group? In particular I am interested in the case where $M$ is a hypergeometric motive as implemented in Magma. I only have access to the characteristic polynomial of the Frobenius operator for any prime $p$, so ideally any characterization would need only this information. Sato-Tate data indicates it is $\mathrm{USp}(4)$.
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3$\begingroup$ Could you explain a little bit more about your setup? Over what field do you work, and what type of motive do you consider (as far as I know, this word has many not necessarily equivalent meanings)? $\endgroup$– R. van Dobben de BruynCommented Apr 4, 2017 at 2:51
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$\begingroup$ Added a bit more information. Let me know more is required. $\endgroup$– Watson LaddCommented Apr 7, 2017 at 1:10
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