Consider the Jacobian $J_C$ of hyperelliptic curve $$C\!: y^2 = x^5 + a$$ over a finite field $\mathbb{F}_p$, where $a \in \mathbb{F}_p^*$, $p \equiv 2 \ (\mathrm{mod} \ 5)$, $p > 2$. Let $\pi \in \mathrm{End}_{\mathbb{F}_p}\!(J_C)$ be the Frobenius endomorphism and let $\alpha \in \mathrm{End}_{\mathbb{F}_{p^4}}\!(J_C)$ be the endomorphism induced by the map $$C \to C,\quad (x, y) \mapsto (\zeta x, y),$$ where $\zeta \in \mathbb{F}_{p^4}$, $\zeta^5 = 1$, $\zeta \neq 1$.

According to the section 4 of "Galbraith, Pujolas, Ritzenhaler, Smith - Distortion maps for supersingular genus 2 curves" the set $$ B = \{\pi^i\alpha^j \mid 0 \leqslant i, j \leqslant 3\} $$ is a basis of $\mathbb{Q}$-vector space $\mathrm{End}^0_{\overline{\mathbb{F}}_p}\!(J_C) = \mathrm{End}_{\overline{\mathbb{F}}_p}\!(J_C) \otimes \mathbb{Q}$.

Denote by $\prime\!: \mathrm{End}^0_{\overline{\mathbb{F}}_p}\!(J_C) \to \mathrm{End}^0_{\overline{\mathbb{F}}_p}\!(J_C)$ the Rosati involution associated to the standard principal polarization $\lambda_C\!: J_C \to \mathrm{Pic}^0(J_C)$.

How to describe the subspace of invariants $$\mathrm{End}^0_{\overline{\mathbb{F}}_p}\!(J_C)^\prime = \{ e \in \mathrm{End}^0_{\overline{\mathbb{F}}_p}\!(J_C) \mid e^\prime = e\}$$ in terms of $B$? In particular, how to express the Verschiebung endomorphism $V = \pi^\prime$ as the linear combination of basis vectors $\pi^i\alpha^j$?


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