For every integer n>10 that is coprime to 6, the curve D_n obtained by dividing the Fermat‑like curve F_n by the symmetric group S_3 has no non‑trivial automorphisms. The proof uses Theorem A together with Torelli’s theorem and shows that every automorphism acts diagonally on a basis of Schur differentials, leaving only the identity. The same argument establishes that the Jacobian of D_n possesses complex multiplication: its endomorphism algebra contains the field generated by a primitive n‑th root of unity.

In the special case n=11, the authors construct an explicit isogeny δ̄ from Jac(D_{11}) to Jac(Y_{11,2}) that is defined over the rational numbers. This map has degree 11^5 and satisfies δ̄^Tδ̄=[11] (where the superscript T denotes the adjoint with respect to the canonical principal polarizations). Its kernel is a maximal isotropic subgroup of Jac(D_{11})[11] that is isomorphic to (ℤ/11ℤ)^5.

The paper does not extend this explicit isogeny to values of n other than 11, so the construction remains limited to that single case. Likewise, the triviality of the automorphism group has been proven only for the specified range of n; values of n not covered by the condition n>10 with gcd(n,6)=1 remain unexamined.

Beyond the curves D_n, the authors analyze related metacyclic quotients C_p for primes p congruent to 1 modulo 3. They prove a lower bound on their gonality: γ(C_p)≥⌈(r+s)/3⌉, where p=r^2+rs+s^2 with r,s>0. They also show that both the genus and gonality of C_p increase without bound as p grows. These results extend previous knowledge about the geometric complexity of such curves.

Together, the work demonstrates a family of curves with exceptionally simple symmetry groups but rich arithmetic structure, and it provides the first explicit isogeny between Jacobians in this setting.