The Galois Group of a Sextic Misiurewicz Polynomial and the Density of Its Root Counts Modulo p

Abstract

Let H(x) = x^6 + 2x^5 + 2x^4 + 2x^3 + x^2 + 1 be the sextic polynomial that parametrizes the primitive (2,3)-Misiurewicz points of the quadratic family f_c(z) = z^2 + c; equivalently, H is the degree-six factor of a_5(c) - a_2(c) (where a_0 = 0, a_{k+1} = a_k^2 + c) remaining after the factors coming from smaller preperiod and period are removed. The number of roots of H modulo the n-th prime is recorded in Sloane's OEIS as sequence A399163. We prove that H is irreducible over Q, has discriminant -2^6 · 743, and that its Galois group G = Gal(H/Q), in its action on the six roots, is the full symmetric group S6; the proof uses Frobenius cycle types at two small primes together with a primitivity argument. By the Chebotarev density theorem this determines the statistics of A399163 completely: the number of roots of H modulo p takes the values 0, 1, 2, 3, 4, 6 with exact densities 53/144, 11/30, 3/16, 1/18, 1/48, 1/720, respectively, and it never takes the value 5. We compare these predictions with the published terms of A399163 and with an independent computation over the first 20,000 primes.

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