# parametrizeConic -- Option whether to rationally parametrize conics.

## Description

If this option is set true and the computation of rParametrizePlaneCurve or parametrize leads to a conic (even degree case) and this conic has a rational point then this conic is also parametrized. So the final result will be a parametrization over \mathbb{P}^{1}. If the conic does not have a rational point a warning is displayed and the parametrization over the conic is returned.

 i1 : K=QQ; i2 : R=K[v,u,z]; i3 : I=ideal(v^8-u^3*(z+u)^5); o3 : Ideal of R i4 : J=ideal(u^6+4*u^5*z+6*u^4*z^2+4*u^3*z^3+u^2*z^4,v*u^5+3*v*u^4*z+3*v*u^3*z^2+v*u^2*z^3,v^2*u^4+3*v^2*u^3*z+3*v^2*u^2*z^2+v^2*u*z^3,v^3*u^3+2*v^3*u^2*z+v^3*u*z^2,v^4*u^2+v^4*u*z,v^5*u+v^5*z,v^6); o4 : Ideal of R i5 : rParametrizePlaneCurve(I,J,parametrizeConic=>true) o5 = | t_0^3t_1^5 | | -t_0^8 | | t_0^8-t_1^8 | 3 1 o5 : Matrix (QQ[t , t ]) <--- (QQ[t , t ]) 0 1 0 1

## Caveat

If rParametrizeConic is changed such that it passes to a degree 2 field extension if the degree of C is even and the conic does not have a rational point, then the result will have entries in the homogeneous coordinate ring of \mathbb{P}^{1} over this extension.