Is there software (preferably open source) besides Magma for:
(1) Deciding if a surface is rational
(2) Compute a parameterization of the surface
And
(3) Relatively easy to code algorithm for (1) or (2).
I found the main paper for (3) `Josef Schicho. Rational Parameterization of Surfaces', but this algorithm doesn't appear easy to code for me.
One can use `IsRational(Scheme)' in the Magma web calculator - it timeouts for complicated examples because of limitation of 60 seconds of execution.
The motivation is just doing few numerical experiments.
EDIT Per M.P.'s request here are the surfaces I am currently interested in (non trivial rational points or parameterization)
2*x^4*y^3*z^3 - x^4*y^2*z^4 + x^2*y^4*z^4 + 2*x^4*y^3*z + 2*x^4*y^2*z^2 - 2*x^2*y^4*z^2 - 2*x^4*y*z^3 + 4*x^2*y^3*z^3 - x^4*y^2 + x^2*y^4 - 2*x^4*y*z + 4*x^2*y^3*z - 4*x^2*y*z^3 + 2*y^3*z^3 + x^2*z^4 - y^2*z^4 - 4*x^2*y*z + 2*y^3*z - 2*x^2*z^2 + 2*y^2*z^2 - 2*y*z^3 + x^2 - y^2 - 2*y*z = 0
Almost certainly the second is not rational:
x^4*y^2*z^4 + x^2*y^4*z^4 - 2*x^4*y^2*z^2 - 2*x^2*y^4*z^2 - 4*x^2*y^2*z^4 + x^4*y^2 + x^2*y^4 - 8*x^2*y^2*z^2 + x^2*z^4 + y^2*z^4 - 4*x^2*y^2 - 2*x^2*z^2 - 2*y^2*z^2 + x^2 + y^2 = 0