Jacobian map C^3->C^3: constant det -2, three distinct points -> (-1/4,0,0)

jc.py

import sympy as sp

x, y, z = sp.symbols('x y z')
u = 1 + x*y
w = 4 + 3*x*y
F = [u**3*z + y**2*u*w, y + 3*x*u**2*z + 3*x*y**2*w, 2*x - 3*x**2*y - x**3*z]

J = sp.Matrix([[sp.diff(f, v) for v in (x, y, z)] for f in F])
print(sp.expand(J.det()))

for p in [(0, 0, sp.Rational(-1, 4)),
          (1, sp.Rational(-3, 2), sp.Rational(13, 2)),
          (-1, sp.Rational(3, 2), sp.Rational(13, 2))]:
    s = dict(zip((x, y, z), p))
    print(p, tuple(sp.simplify(f.subs(s)) for f in F))