Hermitian coverage of the maximal superelliptic curve

Determine whether, for odd integers r and q=p^r with p\ge 5 prime and p\equiv -1\pmod 3, the curve C: y^3=x^{(q^2+1)/2}+x over F_{q^6} is covered over F_{q^6} by the Hermitian curve \mathcal H_{q^3}: Y^{q^3+1}=X^{q^3}+X.

Background

For odd r, the paper proves that the superelliptic curve C: y3=x{(q2+1)/2}+x is maximal over F_{q6} and has genus (q2-1)/2. The same genus occurs among quotients of the Hermitian curve \mathcal H_{q3}, but equality of genera does not establish an isomorphism, a subcover relation, or any covering map between the two curves.

The unresolved question is whether C is covered by \mathcal H_{q3} over the specified base field. A positive answer would connect the explicit Kummer/superelliptic construction to the Hermitian-quotient construction of maximal curves; the paper leaves this question for future investigation.

References

It therefore remains natural to ask whether, for odd $r$, the curve

C:\qquad y3=x{(q2+1)/2}+x

is covered over $F_{q6}$ by the Hermitian curve

\mathcal H_{q3}:\qquad Y{q3+1}=X{q3}+X.

We leave this question for future investigation.

— Maximal and minimal curves of the form $y^3=x^{(q^2+1)/2}+x$  (2608.20654 - Dias et al., 21 Aug 2026) in Section 4, “The genus and Hermitian quotients”