sage: c = Conic(GF(2), [1,1,1,1,1,0])
....: 
sage: c.parametrization()
(Scheme morphism:
   From: Projective Space of dimension 1 over Finite Field of size 2
   To:   Projective Conic Curve over Finite Field of size 2 defined by x^2 + x*y + y^2 + x*z + y*z
   Defn: Defined on coordinates by sending (x : y) to
         (x*y + y^2 : x^2 + x*y : x^2 + x*y + y^2),
 Scheme morphism:
   From: Projective Conic Curve over Finite Field of size 2 defined by x^2 + x*y + y^2 + x*z + y*z
   To:   Projective Space of dimension 1 over Finite Field of size 2
   Defn: Defined on coordinates by sending (x : y : z) to
         (y : x))
sage: f, g = c.parametrization()
sage: (g*f).is_one()
False

sage: R.<x,y,z> = QQ[]                                                                                                                                                              
sage: C = Curve(7*x^2 + 2*y*z + z^2)                                                                                                                                                
sage: f, g = C.parametrization(); f,g                                                                                                                                               
(Scheme morphism:
   From: Projective Space of dimension 1 over Rational Field
   To:   Projective Conic Curve over Rational Field defined by 7*x^2 + 2*y*z + z^2
   Defn: Defined on coordinates by sending (x : y) to
         (-2*x*y : x^2 + 7*y^2 : -2*x^2),
 Scheme morphism:
   From: Projective Conic Curve over Rational Field defined by 7*x^2 + 2*y*z + z^2
   To:   Projective Space of dimension 1 over Rational Field
   Defn: Defined on coordinates by sending (x : y : z) to
         (-1/2*x : 1/7*y + 1/14*z))
sage: (g*f).is_one()                                                                                                                                                                
False
sage: g([0, -1, 2])                                                                                                                                                                 
...
ValueError: [0, 0] does not define a valid point since all entries are 0
sage: p = g.domain().defining_polynomial()                                                                                                                                          
sage: p([0, -1, 2])                                                                                                                                                                 
0