I would like to construct hyperelliptic curves whose Jacobians are isogenous to the square of a supersingular elliptic curve over $\mathbb{F}_{p^2}$

My question is motivated by the following example.

Let $H/\mathbb{F}_{5^2}$ be the hyperelliptic curve given by $y^2 = x^6 + 1$ and $E/\mathbb{F}_{5^2}$ an elliptic curve $y^2 = x^3 + 1$, then we have that $J:=Jac(H)$ has characteristic polynomial of Frobenius given by $(t + 5)^4$.

There is a map $\psi:H\to E$ given by $(x,y)\mapsto (x^2,y)$ and $E$ is supersingular.

If $\phi\in End(J)$ is the $5^2$-Frobenius endomorphism then we have that $\phi = [-5]$

I think this is a direct consequence of $J$ being isogenous to a the square of a supersingular elliptic curve as the Frobenius satifies its characteristic polynomial in the Tate module, but I am not sure if this suffices.

I would like to know if someone can point me out to a characterization of hyperelliptic curves of genus two where the Frobenius in the endomorphism ring of its Jacobian is the $[n]$ map for some $n$.

In fact it will be nice to construct the hyperelliptic curve of genus 2 such that its jacobian is isogenous to the square of a supersingular elliptic curve.

Thanks.

Just for fun, in MAGMA, I have that the curve $H$ is isomorphic to $y^2=\alpha x^5 + \alpha^{13}x$ and the output to check this is using the generic point of $H$ as follows:

Note: I changed the angle brackets by () as I could not find how to show them.

> F(a) := FiniteField(5^2); > P(x) := PolynomialRing(F); > f1 := x^6 + 1; > f := a*x^5 + a^13*x; > H1 := HyperellipticCurve(f1); > H := HyperellipticCurve(f); > IsIsomorphic(H1,H); true Mapping from: CrvHyp: H1 to CrvHyp: H with equations : 2*$.1 + 4*$.3 a^5*$.2 a^8*$.1 + a^22*$.3 and inverse a^22*$.1 + $.3 a^4*$.2 a^20*$.1 + 2*$.3 > FH(X,Y) := FunctionField(H); > Hext := BaseExtend(H,FH); > Jext := Jacobian(Hext); > R(z) := PolynomialRing(FH); > PtJ := Jext![z-X,Y]; > FrPtJ := Jext![z-X^25,Y^25]; > FrPtJ; (x + 4*X^25, (4*X^60 + 2*X^56 + 4*X^52 + 2*X^40 + X^36 + 2*X^32 + 4*X^20 + 2*X^16 + 4*X^12)*Y, 1) > -5*PtJ; (x + 4*X^25, (4*X^60 + 2*X^56 + 4*X^52 + 2*X^40 + X^36 + 2*X^32 + 4*X^20 + 2*X^16 + 4*X^12)*Y, 1) > FrPtJ + 5*PtJ; (1, 0, 0) > w(t) := PolynomialRing(Rationals()); > t^(2*Genus(H))*Evaluate(Numerator(ZetaFunction(H)),1/t); t^4 + 20*t^3 + 150*t^2 + 500*t + 625 > Factorization(t^4 + 20*t^3 + 150*t^2 + 500*t + 625); [ (t + 5, 4) ]