# Is the generalized Kummer threefold rational in characteristics 3?

Let $$E_i\!: y_i^2 = x_i^3 - x_i$$, $$i = 1, 2, 3$$ be three copies of the supersingular elliptic curve in characteristics $$3$$. Consider on $$E_i$$ the following automorphism of order $$3$$: $$\sigma(x_i,y_i) = (x_i+1,y_i).$$

It is obvious that the generalized Kummer surface $$E_1\!\times\! E_2/(\sigma,\sigma)\!: \ y_1^2 - y_2^2 = t^3-t$$ (where $$t = x_1-x_2$$) is rational, because there is a natural conic bundle structure with respect to the variable $$t$$.

Consider the generalized Kummer threefold $$E_1\!\times\! E_2 \!\times\! E_3/(\sigma,\sigma,\sigma)\!: \begin{cases} y_1^2 - y_2^2 = t^3-t,\\ y_1^2 - y_3^2 = s^3-s, \end{cases}$$ where $$t = x_1-x_2$$, $$s = x_1-x_3$$. Is this threefold also rational? For example, is there any bundle structure?