For a prime $p > 3$ consider the quadratic finite field extension $\mathbb{F}_{p^2}/\mathbb{F}_{p}$. Also, consider the elliptic curves $$ E\!: y_0^2 = x_0^3 + ax_0 + b,\qquad E^{(1)}\!: y_1^2 = x_1^3 + a^px_1 + b^p $$ over $\mathbb{F}_{p^2}$. It is well known that there is an $\mathbb{F}_{p^2}$-isomorphism $\varphi\!: E \!\times\! E^{(1)} \to W$, where $W$ is the Weil restriction of $E$ (or $E^{(1)}$) with respect to $\mathbb{F}_{p^2}/\mathbb{F}_{p}$.

Assume that $\mathbb{F}_{p^2} = \mathbb{F}_{p}(\sqrt{\gamma})$, where $\gamma$ is a quadratic non-residue in $\mathbb{F}_{p}$. Besides, let $$ a := a_0 + a_1\sqrt{\gamma},\qquad b := b_0 + b_1\sqrt{\gamma},\qquad x_0 := u_0 + u_1\sqrt{\gamma},\qquad y_0 := v_0 + v_1\sqrt{\gamma}, $$ where $a_0, a_1, b_0, b_1 \in \mathbb{F}_{p}$. Then $W$ obviously has the affine model $$\left\{\begin{array} \ v_0^2 + \gamma v_1^2 = u_0^3 + 3\gamma u_0u_1^2 + a_0u_0 + a_1\gamma u_1 + b_0,\\ 2v_0v_1 = \gamma u_1^3 + 3u_0^2u_1 + a_0u_1 + a_1u_0 + b_1 \end{array}\right. \quad\subset\quad \mathbb{A}^4_{(u_0,u_1,v_0,v_1)} $$

1. How to explicitly find the isomorphism $\varphi$ for the given affine model?

2. If the previous question is very difficult, may be the following one is simpler. The curves $E$, $E^{(1)}$ has the natural involutions $$ [-1]\!: (x_0,y_0) \mapsto (x_0, -y_0),\qquad [-1]^{(1)}\!: (x_1,y_1) \mapsto (x_1, -y_1) $$ respectively. How to explicitly write out their images (under $\varphi$) on the affine model $W \subset \mathbb{A}^4_{(u_0,u_1,v_0,v_1)}$?

3. Is the geometric quotient $W/\langle [-1], [-1]^{(1)} \rangle$ the quadratic surface of Picard $\mathbb{F}_{p}$-number 1, i.e., so-called elliptic quadratic surface?