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?


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