Quartic K3 surfaces with trivial automorphism group

In 2004, Bjorn Poonen proved that for every field k and every integers n ≥ 1, d ≥ 3 with (n,d) not equal to (1,3) and (2,4), there exists a smooth hypersurface X in Pⁿ⁺¹ over k of degree d such that the automorphism group of X over the algebraic closure of k is trivial. For (n,d) = (1,3) such hypersurfaces do not exist. In this case X would be a plane cubic curve and by choosing a flex as the origin, such an X obtains the structure of an elliptic curve on which multiplication by -1 is a nontrivial automorphism. We will deal with the case (n,d) = (2,4) and show that smooth quartic surfaces in P³ with trivial automorphism group do exist if the characteristic of k is at most 5.