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Sep 11, 2018 at 13:17 comment added David Loeffler Of course, this leads immediately to a much harder question: can one always find an $f$ satisfying the conditions of your question such that the mod p Galois representation of f has large image?
Sep 10, 2018 at 20:59 vote accept François Brunault
Sep 10, 2018 at 20:58 comment added François Brunault Very nice and helpful answer, thank you!
Sep 10, 2018 at 15:28 history answered David Loeffler CC BY-SA 4.0