How is this surface related to the square of that CM elliptic curve?

Question

I have come across the following surface: let $X$ be the double covering of $\mathbb{P}_\mathbb{Z}^2$ defined by the equation $$y^2=x_0^6+x_1^6+x_2^6$$ where $y$ is a variable of degree 3.

There is an obvious action of $\mu_6 \times \mu_6$.

Less obvious, I have made some short calculations that show a great link between this surface over $\mathbb{F}_p$ and $E^2$ over $\mathbb{F}_p$, for any prime $p$, where $E$ is the well-known CM elliptic curve $$y^2=x^3+1$$

Without getting too much into specifics, the surface's Brauer group seems to give rise to a modular representation coming from a weight $3$ cusp form, that seems to be a modular form also attached to $E^2$ (in the same way), maybe up to a character (of degree 6). This is just some very simple mod $p$ calculations and strong multiplicity one. But the question isn't about the validity of these calculations, so I don't want you to dwell on this.

The described link makes me guess that the surface is geometrically (and not only arithmetically) related to the squared elliptic curve. Are they birationally equivalent? Maybe after dividing $E^2$ by some subgroup of $\mu_6 \times \mu_6$? How does one begin to check this in sage?

Answer 1

This is probably more complicated then necessary, but anyway:

Let $C_1$ be the smooth genus 2 curve with affine equation $y^2=x^6+1$ and $C_2$ be the genus 10 curve with affine equation $w^6=z^6+1$. There is a natural $\mu_6$-action on $C_1\times C_2$ such that $X$ is birational to the quotient $(C_1\times C_2 )/\mu_6$.

Both curve $C_1$ and $C_2$ admit morphisms to $E$. Therefore the Jacobians of both $C_1$ and $C_2$ are isogenous to nonsimple abelian varieties $A_1,A_2$, which have $E$ as a factor.

This implies that the modular form attached to $E^2$ occurs in the cohomology of $C_1\times C_2$. Probably, this modular form occurs in the $\mu_6$-invariant part of the cohomology of $C_1\times C_2$ (this is easy to check, bit I can't be bothered to check it now). If so then this would be an explanation of the above described phenomena.

Answer 2

This seems to be a K3 surface, and then one would suspect a link with the Kummer surface of $E\times E$. Similar things turn up in a paper of Beukers and Stienstra, "On the Picard-Fuchs equation and the formal Brauer group of certain elliptic K3-surfaces", Math. Ann. 271 (1985), 269-304, and also in a paper of F. Jouve "The geometry of the third moment of exponential sums".