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In this paper we show an explicit polynomial in Q[x] that has Galois group SL2(F16), filling in a gap in the tables of Juergen Klueners and Gunther Malle. The computation of this polynomial uses modular forms and their Galois representations.

A Polynomial with Galois Groups SL2(F16)

A Polynomial with Galois Groups SL₂(F₁₆)