Crossposted from [MSE](https://math.stackexchange.com/questions/1401214/what-is-the-type-of-the-surfaces-x5-y5-z2-x-0-and-x5-y5-z2).

I am interested what is the type of the surfaces over the
rationals
$$ x^5 - y^5 + z^2 + x=0$$

and

$$ x^5 - y^5 + z^2 + x+1=0$$

Magma's `KodairaEnriquesType(S : CheckADE:=true);` fails
to compute it.

According to Magma they are not rational.

Partial answers (e.g. it is not $X$) or approaches
how to compute with a CAS are welcome.