Fermat quotients with CM Jacobians and trivial automorphism group

Let Fₙ be the Fermat curve of degree n over Q, and let Dₙ=Fₙ/S₃ be its quotient by coordinate permutations. We prove that Dₙ has trivial geometric automorphism group for every n≥ 11 with (n,6)=1. Its Jacobian has complex multiplication. For $n=11$, this Jacobian is absolutely simple of dimension five. We construct an isogeny over Q of degree 11⁵ from Jac(D₁₁) to the Jacobian of y¹¹=x(1-x)². For primes p≡ 1 3, we also consider the metacyclic quotients Cₚ of Fₚ over Q. By a theorem of Carocca, Lange and Rodríguez, their Jacobians are absolutely simple and have complex multiplication. We prove that Cₚ has trivial geometric automorphism group if and only if p≥ 19. Write p=r²+rs+s² with $r,s$ positive integers. The natural degree-three cover Yₚ→ Cₚ satisfies gon(Yₚ)=r+s, so gon(Cₚ)≥ (r+s)/3. Since g(Cₚ)=(p-1)/6, both the genus and the gonality of Cₚ tend to infinity with p.