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We consider the distribution of normalized Frobenius traces for two familiesof genus 3 hyperelliptic curves over Q that have large automorphism groups:y^2=x^8+c and y^2=x^7-cx with c in Q*. We give efficient algorithms to computethe trace of Frobenius for curves in these families at primes of goodreduction. Using data generated by these algorithms, we obtain a heuristicdescription of the Sato-Tate groups that arise, both generically and forparticular values of c. We then prove that these heuristic descriptions arecorrect by explicitly computing the Sato-Tate groups via the correspondencebetween Sato-Tate groups and Galois endomorphism types.

Sato-Tate groups of y^2=x^8+c and y^2=x^7-cx