Crossposted from MSE.

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.