#A conic of PG(2,8)
gf := GF(8);
vec := gf^3;
r := PolynomialRing( gf, 3);
poly := r.1^2 + r.2 * r.3;
form := QuadraticFormByPolynomial( poly, r );
Display( form );
IsDegenerateForm( form );
IsSingularForm( form );
WittIndex( form );
IsParabolicForm( form );
RadicalOfForm( form );
canonical := IsometricCanonicalForm( form );
form = canonical;
go := GO(3,8);
mat := InvariantQuadraticForm( go )!.matrix;
gapform := QuadraticFormByMatrix( mat, GF(8) );
b := BaseChangeToCanonical( gapform );
hom := BaseChangeHomomorphism( b, GF(8) );
newgo := Image(hom, go);
conic := Filtered(vec, x -> IsZero( x^form ));;
Size(conic);
orbs := Orbits(newgo, conic, OnRight);;
List(orbs,Size);
quit;
