// (192,181) signature (2,3,8)

K<z_8>:=CyclotomicField(8);
i:=z_8^2;
sqrt2:=z_8-z_8^3;
P4<x_0,x_1,x_2,x_3,x_4>:=ProjectiveSpace(K,4);
X:=Scheme(P4,[
x_0^2+x_3^2+x_4^2,
x_1^2+x_3^2-x_4^2,
x_2^2+x_3*x_4
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[0, 0, 1/2*(i + 1), 0, 0],
[0, -1, 0, 0, 0],
[1-i, 0, 0, 0, 0],
[0, 0, 0, -1/sqrt2, -i/sqrt2],
[0, 0, 0, i/sqrt2, 1/sqrt2]
]);
B:=Matrix([
[0, z_8^-1, 0, 0, 0],
[0, 0, -1/sqrt2, 0, 0],
[-1-i, 0, 0, 0, 0],
[0, 0, 0, 1/2*(i - 1), 1/2*(-i - 1)],
[0, 0, 0, 1/2*(1-i), 1/2*(-i - 1)]
]);
Order(A);
Order(B);
Order( (A*B)^(-1));
GL5K:=GeneralLinearGroup(5,K);
IdentifyGroup(sub<GL5K | A,B>);
Automorphism(X,A);
Automorphism(X,B);


// (160,234) signature (2,4,5)
K<z_5>:=CyclotomicField(5);
P4<x_0,x_1,x_2,x_3,x_4>:=ProjectiveSpace(K,4);
X:=Scheme(P4,[
x_0^2+x_1^2+x_2^2+x_3^2+x_4^2,
x_0^2+z_5*x_1^2+z_5^2*x_2^2+z_5^3*x_3^2+z_5^4*x_4^2,
z_5^4*x_0^2+z_5^3*x_1^2+z_5^2*x_2^2+z_5*x_3^2+x_4^2
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[0,0,0,-1,0],
[0,0,1,0,0],
[0,1,0,0,0],
[-1,0,0,0,0],
[0,0,0,0,-1]
]);
B:=Matrix([
[-1,0,0,0,0],
[0,0,0,0,-1],
[0,0,0,1,0],
[0,0,-1,0,0],
[0,1,0,0,0]
]);
Order(A);
Order(B);
Order( (A*B)^(-1));
GL5K:=GeneralLinearGroup(5,K);
IdentifyGroup(sub<GL5K | A,B>);
Automorphism(X,A);
Automorphism(X,B);


// (96,195) signature (2,4,6)
K<z_3>:=CyclotomicField(3);
P4<x_0,x_1,x_2,x_3,x_4>:=ProjectiveSpace(K,4);
X:=Scheme(P4,[
x_0^2 + x_3^2 + x_4^2,
x_1^2+z_3*x_3^2+z_3^2*x_4^2,
x_2^2+z_3^2*x_3^2+z_3*x_4^2
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[0,0,-1,0,0],
[0,-1,0,0,0],
[-1,0,0,0,0],
[0,0,0,0,z_3],
[0,0,0,z_3^2,0]
]);
B:=Matrix([
[-1,0,0,0,0],
[0,0,-1,0,0],
[0,1,0,0,0],
[0,0,0,0,1],
[0,0,0,-1,0]
]);
Order(A);
Order(B);
Order( (A*B)^(-1));
GL5K:=GeneralLinearGroup(5,K);
IdentifyGroup(sub<GL5K | A,B>);
Automorphism(X,A);
Automorphism(X,B);


// (64,32) signature (2,4,8)
K<i>:=CyclotomicField(4);
P4<x_0,x_1,x_2,x_3,x_4>:=ProjectiveSpace(K,4);
X:=Scheme(P4,[
x_0^2+x_1^2+x_2^2+x_3^2+x_4^2,
x_0^2+i*x_1^2-x_2^2-i*x_3^2,
x_0^2-x_1^2+x_2^2-x_3^2
]);
IsSingular(X);
Dimension(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[-1,0,0,0,0],
[0,1,0,0,0],
[0,0,-1,0,0],
[0,0,0,-1,0],
[0,0,0,0,-1]
]);
B:=Matrix([
[0,0,0,-i,0],
[i,0,0,0,0],
[0,-i,0,0,0],
[0,0,i,0,0],
[0,0,0,0,i]
]);
Order(A);
Order(B);
Order( (A*B)^(-1));
GL5K:=GeneralLinearGroup(5,K);
IdentifyGroup(sub<GL5K | A,B>);
Automorphism(X, A);
Automorphism(X, B);

// (30,2) signature (2,6,15)
K<z_15>:=CyclotomicField(15);
P4<x_0,x_1,x_2,x_3,x_4>:=ProjectiveSpace(K,4);
X:=Scheme(P4,[
x_0*x_3-x_1*x_2,
x_0*x_4-x_1*x_3,
x_2*x_4-x_3^2,
x_0^2*x_1-x_3*x_4^2+x_2^3,
x_0*x_1^2-x_4^3+x_2^2*x_3
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[0,z_15^12,0,0,0],
[z_15^3,0,0,0,0],
[0,0,0,0,-z_15^9],
[0,0,0,-1,0],
[0,0,-z_15^6,0,0]
]);
B:=Matrix([
[0,z_15^11,0,0,0],
[z_15^14,0,0,0,0],
[0,0,0,0,-z_15^7],
[0,0,0,-z_15^10,0],
[0,0,-z_15^13,0,0]
]);
Order(A);
Order(B);
Order((A*B)^(-1));
GL5K:=GeneralLinearGroup(5,K);
IdentifyGroup(sub<GL5K | A,B>);
Automorphism(X,A);
Automorphism(X,B);


// (48,48) signature (2,2,2,3)
K:=RationalField();
P4<x_0,x_1,x_2,x_3,x_4>:=ProjectiveSpace(K,4);
c_4:=17;
X:=Scheme(P4,[
x_0^2 - x_0*x_1 + x_1^2+x_2^2 + x_2*x_4 + x_3^2 - x_3*x_4 + x_4^2,
x_0^2 - x_1^2+c_4*(x_2^2 - 2*x_2*x_3 + 2*x_2*x_4 + x_3^2 - 2*x_3*x_4),
x_0*x_1 - 1/2*x_1^2 + c_4*(-2*x_2*x_3 + x_2*x_4 - x_3*x_4 + 1/2*x_4^2)
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[-1, 0, 0, 0, 0],
[ 0, -1, 0, 0, 0],
[ 0, 0, -1, 0, 0],
[ 0, 0, 0, -1, 0],
[ 0, 0, -1, 1, 1]
]);
B:=Matrix([
[-1, -1, 0, 0, 0],
[ 0, 1, 0, 0, 0],
[ 0, 0, 0, 0, -1],
[ 0, 0, 1, -1, -1],
[ 0, 0, -1, 0, 0]
]);
C:=Matrix([
[-1, 0, 0, 0, 0],
[ 1, 1, 0, 0, 0],
[ 0, 0, -1, 0, 0],
[ 0, 0, 0, 1, 0],
[ 0, 0, 0, -1, -1]
]);
Order(A);
Order(B);
Order(C);
Order((A*B*C)^(-1));
GL5K:=GeneralLinearGroup(5,K);
IdentifyGroup(sub<GL5K | A,B,C>);
Automorphism(X,A);
Automorphism(X,B);
Automorphism(X,C);

// (32,43) signature (2,2,2,4)
K<i>:=CyclotomicField(4);
P4<x_0,x_1,x_2,x_3,x_4>:=ProjectiveSpace(K,4);
c_4:=17+i;
X:=Scheme(P4,[x_0^2+x_1*x_2 - i*x_3*x_4,
x_1^2+i*x_3^2+c_4*(x_2^2-i*x_4^2),
i*x_2^2-x_4^2+c_4*(i*x_1^2+x_3^2)
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[-1,0,0,0,0],
[0,0,i,0,0],
[0,-i,0,0,0],
[0,0,0,0,-i],
[0,0,0,i,0]
]);
B:=Matrix([
[-1,0,0,0,0],
[0,0,0,0,-i],
[0,0,0,-1,0],
[0,0,-1,0,0],
[0,i,0,0,0]
]);
C:=Matrix([
[-1,0,0,0,0],
[0,0,-i,0,0],
[0,i,0,0,0],
[0,0,0,0,-i],
[0,0,0,i,0]
]);
Order(A);
Order(B);
Order(C);
Order( (A*B*C)^(-1));
GL5K:=GeneralLinearGroup(5,K);
IdentifyGroup(sub<GL5K | A,B,C>);
Automorphism(X,A);
Automorphism(X,B);
Automorphism(X,C);


// (32,28) signature (2,2,2,4)
K<i>:=CyclotomicField(4);
P4<x_0,x_1,x_2,x_3,x_4>:=ProjectiveSpace(K,4);
c_6:=1;
X:=Scheme(P4,[
x_0^2+x_1^2 + x_2^2,
x_1*x_2+x_3^2 - x_4^2,
x_1^2 - x_2^2+c_6*(x_3*x_4)
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[-1, 0, 0, 0, 0],
[0, -1, 0, 0, 0],
[0, 0, -1, 0, 0],
[0, 0, 0, 0, -i],
[0, 0, 0, i, 0]
]);
B:=Matrix([
[-1, 0, 0, 0, 0],
[0, -1, 0, 0, 0],
[0, 0, 1, 0, 0],
[0, 0, 0, 0, -1],
[0, 0, 0, -1, 0]
]);
C:=Matrix([
[-1, 0, 0, 0, 0],
[0, 0, -1, 0, 0],
[0, -1, 0, 0, 0],
[0, 0, 0, -1, 0],
[0, 0, 0, 0, 1]
]);
Order(A);
Order(B);
Order(C);
Order( (A*B*C)^(-1));
GL5K:=GeneralLinearGroup(5,K);
IdentifyGroup(sub<GL5K | A,B,C>);
Automorphism(X,A);
Automorphism(X,B);
Automorphism(X,C);

// (32,27) signature (2,2,2,4)
K<i>:=CyclotomicField(4);
P4<x_0,x_1,x_2,x_3,x_4>:=ProjectiveSpace(K,4);
c_8:=1;
X:=Scheme(P4,[
x_0^2+x_3^2 + x_4^2,
x_1*x_2+ x_3^2 + x_4^2,
x_1^2 + x_2^2+c_8*x_3*x_4
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[-1, 0, 0, 0, 0],
[0, -1, 0, 0, 0],
[0, 0, -1, 0, 0],
[0, 0, 0, 0, -1],
[0, 0, 0, -1, 0]
]);
B:=Matrix([
[-1, 0, 0, 0, 0],
[0, 0, -1, 0, 0],
[0, -1, 0, 0, 0],
[0, 0, 0, -1, 0],
[0, 0, 0, 0, -1]
]);
C:=Matrix([
[-1, 0, 0, 0, 0],
[0, 0, -i, 0, 0],
[0, i, 0, 0, 0],
[0, 0, 0, -1, 0],
[0, 0, 0, 0, 1]
]);
Order(A);
Order(B);
Order(C);
Order( (A*B*C)^(-1));
GL5K:=GeneralLinearGroup(5,K);
IdentifyGroup(sub<GL5K | A,B,C>);
Automorphism(X,A);
Automorphism(X,B);
Automorphism(X,C);


// (24,8) signature (2,2,2,6)
K<z_6>:=CyclotomicField(6);
P4<x_0,x_1,x_2,x_3,x_4>:=ProjectiveSpace(K,4);
c_4:=1;
X:=Scheme(P4,[
x_0^2 + x_1^2 + x_1*x_2 + x_2^2,
x_0*x_1+c_4*(x_1^2 - 2*x_1*x_2 - 2*x_2^2)+x_3^2 + x_4^2,
x_0*x_2+c_4*(-2*x_1^2 - 2*x_1*x_2 + x_2^2)-z_6*x_3^2 + (z_6 - 1)*x_4^2
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[-1,0,0,0,0],
[0,-1,0,0,0],
[0,0,-1,0,0],
[0,0,0,-1,0],
[0,0,0,0,1]
]);
B:=Matrix([
[-1,0,0,0,0],
[0,0,-1,0,0],
[0,-1,0,0,0],
[0,0,0,0,z_6-1],
[0,0,0,-z_6,0]
]);
C:=Matrix([
[-1,0,0,0,0],
[0,1,0,0,0],
[0,1,-1,0,0],
[0,0,0,0,z_6],
[0,0,0,-z_6+1,0]
]);
Order(A);
Order(B);
Order(C);
Order( (A*B*C)^(-1));
GL5K:=GeneralLinearGroup(5,K);
IdentifyGroup(sub<GL5K | A,B,C>);
Automorphism(X,A);
Automorphism(X,B);
Automorphism(X,C);



// (24,13) signature (2,2,3,3)
K<z_6>:=CyclotomicField(6);
P4<x_0,x_1,x_2,x_3,x_4>:=ProjectiveSpace(K,4);
c_6:=17;
X:=Scheme(P4,[
x_0*x_1 + x_2^2 + x_3^2 + x_4^2,
x_1^2 + x_2^2 + (z_6 - 1)*x_3^2 - z_6*x_4^2,
x_0^2 + c_6*(x_2^2 - z_6*x_3^2 + (z_6-1)*x_4^2)
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[-1,0,0,0,0],
[0,-1,0,0,0],
[0,0,1,0,0],
[0,0,0,-1,0],
[0,0,0,0,-1]
]);
B:=Matrix([
[-1,0,0,0,0],
[0,-1,0,0,0],
[0,0,1,0,0],
[0,0,0,-1,0],
[0,0,0,0,-1]
]);
C:=Matrix([
[z_6-1,0,0,0,0],
[0,-z_6,0,0,0],
[0,0,0,-1,0],
[0,0,0,0,1],
[0,0,-1,0,0]
]);
Order(A);
Order(B);
Order(C);
Order( (A*B*C)^(-1));
GL5K:=GeneralLinearGroup(5,K);
IdentifyGroup(sub<GL5K | A,B,C>);
Automorphism(X,A);
Automorphism(X,B);
Automorphism(X,C);