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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 

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