// (504,156) signature (2,3,7)
K<z_7>:=CyclotomicField(7);
P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
X:=Scheme(P6,[
x_0^2+x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2,
x_0^2+z_7*x_1^2+z_7^2*x_2^2+z_7^3*x_3^2+z_7^4*x_4^2+z_7^5*x_5^2+z_7^6*x_6^2,
x_0^2+z_7^(-1)*x_1^2+z_7^(-2)*x_2^2+z_7^(-3)*x_3^2+z_7^(-4)*x_4^2+z_7^(-5)*x_5^2+z_7^(-6)*x_6^2,
(z_7^(-3)-z_7^3)*x_0*x_6-(z_7^(-2)-z_7^2)*x_1*x_4+(z_7-z_7^(-1))*x_3*x_5,
(z_7^(-3)-z_7^3)*x_1*x_0-(z_7^(-2)-z_7^2)*x_2*x_5+(z_7-z_7^(-1))*x_4*x_6,
(z_7^(-3)-z_7^3)*x_2*x_1-(z_7^(-2)-z_7^2)*x_3*x_6+(z_7-z_7^(-1))*x_5*x_0,
(z_7^(-3)-z_7^3)*x_3*x_2-(z_7^(-2)-z_7^2)*x_4*x_0+(z_7-z_7^(-1))*x_6*x_1,
(z_7^(-3)-z_7^3)*x_4*x_3-(z_7^(-2)-z_7^2)*x_5*x_1+(z_7-z_7^(-1))*x_0*x_2,
(z_7^(-3)-z_7^3)*x_5*x_4-(z_7^(-2)-z_7^2)*x_6*x_2+(z_7-z_7^(-1))*x_1*x_3,
(z_7^(-3)-z_7^3)*x_6*x_5-(z_7^(-2)-z_7^2)*x_0*x_3+(z_7-z_7^(-1))*x_2*x_4
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=DiagonalMatrix([1,-1,-1,-1,1,1,-1]);
B:=Matrix([
[0, 1/2, 1/2, -1/2, 0, -1/2, 0],
[-1/2, -1/2, 1/2, 0, -1/2, 0, 0],
[1/2, -1/2, 0, -1/2, 0, 0, -1/2],
[-1/2, 0, -1/2, 0, 0, -1/2, -1/2],
[0, 1/2, 0, 0, -1/2, 1/2, -1/2],
[1/2, 0, 0, 1/2, -1/2, -1/2, 0],
[0, 0, 1/2, 1/2, 1/2, 0, -1/2]
]);
Order(A);
Order(B);
Order( (A*B)^-1);
GL7K:=GeneralLinearGroup(7,K);
IdentifyGroup(sub<GL7K | A,B>);
Automorphism(X,A);
Automorphism(X,B);



// (144,127) signature (2,3,12)
K<z_12>:=CyclotomicField(12);
z_6:=z_12^2;
i:=z_12^3;
z_3:=z_12^4;
P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
X:=Scheme(P6,[x_0^2 + x_3*x_4 - z_6*x_3*x_5 - z_6*x_5*x_6,
2*i*x_1^2 + x_3*x_4 + z_6*x_3*x_5 + 2*x_4*x_6 - z_6*x_5*x_6,
2*i*x_1*x_2 + (-2*z_6 + 1)*x_3*x_4 + z_6*x_3*x_5 - 2*z_6*x_4*x_6 + z_6*x_5*x_6,
2*i*x_2^2 -x_3*x_4 + (-z_6 + 2)*x_3*x_5 + (2*z_6 - 2)*x_4*x_6 + (-z_6 + 2)*x_5*x_6,
x_1*x_3 - z_6*x_2*x_6 + z_12*x_4^2 + (z_12^3 - 2*z_12)*x_4*x_5 + z_12*x_5^2,
x_1*x_4 + (-z_6 + 1)*x_2*x_5 -x_3*x_6 - x_6^2,
x_1*x_5 - x_2*x_5 + x_3^2 + (-z_6 + 2)*x_3*x_6,
x_1*x_6 + x_2*x_6  + -z_12*x_4^2 + z_12*x_4*x_5 - z_12*x_5^2,
x_2*x_3 - z_6*x_2*x_6 + z_12*x_4^2,
x_2*x_4 + (-z_6 - 1)*x_2*x_5 + z_6*x_3^2 + 2*z_6*x_3*x_6 + z_6*x_6^2
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[-1,0,0,0,0,0,0],
[0,z_12^-1,-z_12, 0,0,0,0],
[0,-z_12,-z_12^-1,0,0,0,0],
[0,0,0,0,z_12^2,z_12^2,0],
[0,0,0,-z_3,0,0,z_3],
[0,0,0,0,0,0,-z_3],
[0,0,0,0,0,z_6,0]
]);
B:=Matrix([
[z_3,0,0,0,0,0,0],
[0,-1,z_3,0,0,0,0],
[0,z_6,0,0,0,0,0],
[0,0,0,z_3,0,0,1],
[0,0,0,0,z_3,0,0],
[0,0,0,0,-z_3,-z_6,0],
[0,0,0,0,0,0,1]
]);
Order(A);
Order(B);
Order( (A*B)^-1);
GL7K:=GeneralLinearGroup(7,K);
IdentifyGroup(sub<GL7K | A,B>);
Automorphism(X,A);
Automorphism(X,B);


// (64,41) signature (2,4,16)
K<z_16>:=CyclotomicField(16);
z_8:=z_16^2;
i:=z_16^4;
P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
X:=Scheme(P6,[
x_0^2+x_1*x_2,
x_3*x_4-z_8*x_5*x_6,
x_1^2+x_3*x_6+i*x_4*x_5,
x_2^2+i*x_3*x_6+x_4*x_5,
x_0*x_1+z_16^7*x_3^2-z_16^5*x_5^2,
x_0*x_2-z_16^7*x_4^2+z_16^5*x_6^2,
x_0*x_3-z_16*x_2*x_6,
x_0*x_4+z_16*x_1*x_5,
x_0*x_5+z_16^7*x_2*x_4,
x_0*x_6-z_16^7*x_1*x_3
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[-1, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 0, 1, 0]
]);
B:=Matrix([
[z_8^2, 0, 0, 0, 0, 0, 0],
[0, 0, -z_8, 0, 0, 0, 0],
[0, -z_8^3, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 0, -z_8, 0],
[0, 0, 0, 0, -z_8^3, 0, 0],
[0, 0, 0, 1, 0, 0, 0]
]);
Order(A);
Order(B);
Order( (A*B)^-1);
GL7K:=GeneralLinearGroup(7,K);
IdentifyGroup(sub<GL7K | A,B>);
Automorphism(X,A);
Automorphism(X,B);

// (54,6) signature (2,6,9)
K<z_18>:=CyclotomicField(18);
z_9:=z_18^2;
z_6:=z_18^3;
z_3:=z_18^6;
P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
X:=Scheme(P6,[
x_1*x_6 + x_2*x_4 + x_3*x_5,
x_0^2-x_1*x_6 +z_6*x_2*x_4 -z_3*x_3*x_5,
x_1*x_4 + z_3*x_2*x_5 - z_6*x_3*x_6,
x_1*x_5 + z_3*x_2*x_6 - z_6*x_3*x_4,
x_0*x_1-z_6*x_5^2-x_4*x_6,
x_0*x_2+x_6^2-z_3*x_4*x_5,
x_0*x_3+z_3*x_4^2+z_6*x_5*x_6,
x_0*x_4-x_1^2+z_3*x_2*x_3,
x_0*x_5-z_3*x_2^2-z_6*x_1*x_3,
x_0*x_6+z_6*x_3^2+x_1*x_2]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[-1,0,0,0,0,0,0],
[0,0,0,0,0,0,z_9^4],
[0,0,0,0,z_9,0,0],
[0,0,0,0,0,z_9^7,0],
[0,0,z_9^-1,0,0,0,0],
[0,0,0,z_9^2,0,0,0],
[0,z_9^5,0,0,0,0,0]
]);
B:=Matrix([
[z_6,0,0,0,0,0,0],
[0,0,0,0,0,z_3,0],
[0,0,0,0,0,0,z_3],
[0,0,0,0,z_3,0,0],
[0,z_3^2,0,0,0,0,0],
[0,0,z_3^2,0,0,0,0],
[0,0,0,z_3^2,0,0,0]
]);
Order(A);
Order(B);
Order( (A*B)^-1);
GL7K:=GeneralLinearGroup(7,K);
IdentifyGroup(sub<GL7K | A,B>);
Automorphism(X,A);
Automorphism(X,B);



// (54,3) signature (2,6,9) cyclic trigonal
K<z_9>:=CyclotomicField(9);
z_3:=z_9^3;
P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
X:=Scheme(P6,[
-x_1*x_2+x_0*x_3,
-x_1*x_3+x_0*x_4,
-x_3^2+x_2*x_4,
-x_1*x_4+x_0*x_5,
-x_3*x_4+x_2*x_5,
-x_4^2+x_3*x_5,
-x_1*x_5+x_0*x_6,
-x_3*x_5+x_2*x_6,
-x_4*x_5+x_3*x_6,
-x_5^2+x_4*x_6,
x_0^3-x_6^2*x_3+x_2^3,
x_0^2*x_1-x_6^2*x_4+x_2^2*x_3,
x_0*x_1^2-x_6^2*x_5+x_2^2*x_4,
x_1^3-x_6^2*x_6+x_2^2*x_5
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[0,1,0,0,0,0,0],
[1,0,0,0,0,0,0],
[0,0,0,0,0,0,-1],
[0,0,0,0,0,-1,0],
[0,0,0,0,-1,0,0],
[0,0,0,-1,0,0,0],
[0,0,-1,0,0,0,0]
]);
B:=Matrix([
[0,z_9^2,0,0,0,0,0],
[z_9,0,0,0,0,0,0],
[0,0,0,0,0,0,-z_9^5],
[0,0,0,0,0,-z_9^4,0],
[0,0,0,0,-z_9^3,0,0],
[0,0,0,-z_9^2,0,0,0],
[0,0,-z_9,0,0,0,0]
]);
Order(A);
Order(B);
Order( (A*B)^-1);
GL7K:=GeneralLinearGroup(7,K);
IdentifyGroup(sub<GL7K | A,B>);
Automorphism(X,A);
Automorphism(X,B);

// (48,32)
K<z_6>:=CyclotomicField(6);
z_3:=z_6^2;
t:=2*z_6 - 1;
P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
X:=Scheme(P6,[
x_0^2+x_3*x_5 + z_6*x_3*x_6 + (-z_6 + 1)*x_4*x_6,
t*(x_1*x_3 - x_2*x_4+ x_1*x_5) - x_1*x_6 + x_2*x_5 - x_2*x_6,
t*(2*x_1*x_4 - x_2*x_5 + x_2*x_6)+ x_1*x_5 - x_1*x_6,
t*(2*x_2*x_3 + x_1*x_6 - x_2*x_5)  + 3*x_1*x_5 + x_2*x_6,
-3*(x_1^2+x_3*x_5 - x_3*x_6) + t*(x_4*x_5 - x_4*x_6)+2*x_5^2 + 2*(z_6 - 1)*x_5*x_6 -2*z_6*x_6^2,
-3*(x_1*x_2 - x_4*x_5)+t*(x_3*x_5 - x_3*x_6 + x_4*x_6-x_5^2) + 2*z_6*x_5*x_6 - x_6^2,
-3*(x_2^2-x_3*x_5 - x_4*x_6)+ t*(x_3*x_6 -3*x_4*x_5) +2*(z_6 + 1)*x_5*x_6 + 2*(z_6 - 1)*x_6^2,
-3*(2*x_4^2-x_3*x_5 + x_3*x_6) +t*(- x_4*x_5 + x_4*x_6)+2*x_5^2 + 2*(z_6 - 1)*x_5*x_6 - 2*z_6*x_6^2,
-3*(-2*x_3*x_4 + x_4*x_5) + t*(- x_3*x_5 + x_3*x_6 - x_4*x_6-x_5^2) + 2*z_6*x_5*x_6 - x_6^2,
-3*(2*x_3^2 + x_3*x_5 +x_4*x_6) +t*(- x_3*x_6 + 3*x_4*x_5) +2*(z_6 + 1)*x_5*x_6 + 2*(z_6 - 1)*x_6^2
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[z_3,0,0,0,0,0,0],
[0,0,-z_3,0,0,0,0],
[0,-z_6,-1,0,0,0,0],
[0,0,0,-1,z_6,0,0],
[0,0,0,z_3,0,0,0],
[0,0,0,0,0,z_3,-z_6],
[0,0,0,0,0,0,1]
]);
B:=Matrix([
[-1,0,0,0,0,0,0],
[0,0,1,0,0,0,0],
[0,-1,0,0,0,0,0],
[0,0,0,0,-1,0,0],
[0,0,0,1,0,0,0],
[0,0,0,0,0,z_3,-z_6],
[0,0,0,0,0,-z_6,-z_3]
]);
Order(A);
Order(B);
Order( (A*B)^-1);
GL7K:=GeneralLinearGroup(7,K);
IdentifyGroup(sub<GL7K | A,B>);
Automorphism(X,A);
Automorphism(X,B);


// (42,4) signature (2,6,21) cyclic trigonal
K<z_21>:=CyclotomicField(21);
z_3:=z_21^7;
z_7:=z_21^3;
P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
X:=Scheme(P6,[
-x_1*x_2+x_0*x_3,
-x_1*x_3+x_0*x_4,
-x_3^2+x_2*x_4,
-x_1*x_4+x_0*x_5,
-x_3*x_4+x_2*x_5,
-x_4^2+x_3*x_5,
-x_1*x_5+x_0*x_6,
-x_3*x_5+x_2*x_6,
-x_4*x_5+x_3*x_6,
-x_5^2+x_4*x_6,
x_0^3-x_6^2*x_2+x_2^2*x_3,
x_0^2*x_1-x_6^2*x_3+x_2^2*x_4,
x_0*x_1^2-x_6^2*x_4+x_2^2*x_5,
x_1^3-x_6^2*x_5+x_2^2*x_6
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[0,z_7^-1,0,0,0,0,0],
[z_7,0,0,0,0,0,0],
[0,0,0,0,0,0,-z_7^3],
[0,0,0,0,0,-z_7^-2,0],
[0,0,0,0,-1,0,0],
[0,0,0,-z_7^2,0,0,0],
[0,0,-z_7^4,0,0,0,0]
]);
B:=Matrix([
[0,z_21^-5,0,0,0,0,0],
[z_21^-2,0,0,0,0,0,0],
[0,0,0,0,0,0,-z_21^8],
[0,0,0,0,0,-z_21^11,0],
[0,0,0,0,z_3+1,0,0],
[0,0,0,-z_21^-4,0,0,0],
[0,0,-z_21^-1,0,0,0,0]
]);
Order(A);
Order(B);
Order( (A*B)^-1);
GL7K:=GeneralLinearGroup(7,K);
IdentifyGroup(sub<GL7K | A,B>);
Automorphism(X,A);
Automorphism(X,B);


// (32,11)
K<z_8>:=CyclotomicField(8);
i:=z_8^2;
P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
X:=Scheme(P6,[x_3*x_5+x_4*x_6,
x_0^2+x_1*x_5+z_8^2*x_2*x_6,
x_1*x_4+z_8^2*x_2*x_3+x_5*x_6,
x_1*x_2+x_3*x_4,
x_1*x_6+z_8^3*x_4*x_5,
x_2*x_5+z_8*x_3*x_6,
x_1^2-i*x_3^2-z_8^3*x_5^2,
x_2^2+i*x_4^2+z_8^3*x_6^2,
-i*x_2*x_4+z_8^3*x_3^2,
x_1*x_3-z_8^3*x_4^2]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[-1,0,0,0,0,0,0],
[0,0,-1,0,0,0,0],
[0,1,0,0,0,0,0],
[0,0,0,0,-i,0,0],
[0,0,0,-i,0,0,0],
[0,0,0,0,0,0,i],
[0,0,0,0,0,i,0]
]);
B:=Matrix([
[i,0,0,0,0,0,0],
[0,1,0,0,0,0,0],
[0,0,i,0,0,0,0],
[0,0,0,-1,0,0,0],
[0,0,0,0,-i,0,0],
[0,0,0,0,0,-1,0],
[0,0,0,0,0,0,i]
]);
Order(A);
Order(B);
Order( (A*B)^-1);
GL7K:=GeneralLinearGroup(7,K);
IdentifyGroup(sub<GL7K | A,B>);
Automorphism(X,A);
Automorphism(X,B);


// (32,10)
K<z_16>:=CyclotomicField(16);
P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
X:=Scheme(P6,[
x_1*x_6+z_16^6*x_2*x_5+x_3*x_4,
x_1*x_2+x_5*x_6,
x_0^2+x_1*x_6-z_16^6*x_2*x_5,
x_3*x_6-z_16^4*x_4*x_5,
x_1^2-z_16^7*x_4^2-z_16^6*x_5^2,
x_2^2+z_16^3*x_3^2-z_16^10*x_6^2,
-z_16^2*x_2*x_6+(z_16^16+z_16^8)*x_4^2-z_16^7*x_5^2,
x_1*x_5+(-z_16^12-z_16^4)*x_3^2-z_16^11*x_6^2,
x_1*x_3+z_16^7*x_4*x_6,
x_2*x_4+z_16*x_3*x_5
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[-1,0,0,0,0,0,0],
[0,0,-1,0,0,0,0],
[0,1,0,0,0,0,0],
[0,0,0,0,-z_16^2,0,0],
[0,0,0,-z_16^6,0,0,0],
[0,0,0,0,0,0,-z_16^6],
[0,0,0,0,0,-z_16^2,0]
]);
B:=Matrix([
[z_16^4,0,0,0,0,0,0],
[0,0,-z_16^6,0,0,0,0],
[0,-z_16^2,0,0,0,0,0],
[0,0,0,0,-z_16^4,0,0],
[0,0,0,z_16^4,0,0,0],
[0,0,0,0,0,0,-z_16^4],
[0,0,0,0,0,-z_16^4,0]
]);
Order(A);
Order(B);
Order( (A*B)^-1);
GL7K:=GeneralLinearGroup(7,K);
IdentifyGroup(sub<GL7K | A,B>);
Automorphism(X,A);
Automorphism(X,B);



// (48,48) signature (2,2,2,4)
P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(RationalField(),6);
c_13:=1;
X:=Scheme(P6,[
x_0^2 + x_4^2 + x_4*x_6 + x_5^2 - x_5*x_6 + x_6^2,
x_1^2 - x_1*x_2 + x_2^2 - x_2*x_3 + x_3^2+x_4^2 + x_4*x_6 + x_5^2 - x_5*x_6 + x_6^2,
(c_13^2+6)*(x_1^2 - 2*x_1*x_3 - x_2^2 + x_3^2)+c_13^2*(x_4^2 - 2*x_4*x_5 + 2*x_4*x_6 + x_5^2 - 2*x_5*x_6),
(c_13^2+6)*(x_1*x_2 - 2*x_1*x_3 - 1/2*x_2^2 + x_2*x_3)+ c_13^2*(-2*x_4*x_5 + x_4*x_6 - x_5*x_6 + 1/2*x_6^2),
c_13*(x_0*x_4 + x_0*x_5 + x_0*x_6)+((c_13^2+6)/2)*(x_1^2 - x_3^2)+(c_13^2/2)*(x_4^2 + 2*x_4*x_6 - x_5^2 + x_6^2),
c_13*(x_0*x_4 + x_0*x_5)+((c_13^2+6)/2)*(x_1*x_2 - x_2*x_3)+(c_13^2/2)*(x_4*x_6 - x_5*x_6 + x_6^2),
c_13*(2*x_0*x_4 + x_0*x_6)+((c_13^2+6)/2)*(x_2^2 - 2*x_2*x_3)+(c_13^2/2)*(2*x_4*x_6 + x_6^2),
x_0*x_1 + x_0*x_3+c_13*(x_1*x_5 - x_3*x_4 - x_3*x_6),
2*x_0*x_1 - x_0*x_2+c_13*(x_1*x_6 + x_2*x_5 - x_2*x_6),
x_0*x_2 - 2*x_0*x_3+c_13*(x_2*x_4 + x_3*x_6)
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[-1, 0, 0, 0, 0, 0, 0],
[ 0, 0, 1, 1, 0, 0, 0],
[ 0, 0, -1, 0, 0, 0, 0],
[ 0, 1, 1, 0, 0, 0, 0],
[ 0, 0, 0, 0, -1, 1, 1],
[ 0, 0, 0, 0, 0, 0, 1],
[ 0, 0, 0, 0, 0, 1, 0]
]);
B:=Matrix([
[-1, 0, 0, 0, 0, 0, 0],
[ 0, 1, 1, 0, 0, 0, 0],
[ 0, 0, -1, 0, 0, 0, 0],
[ 0, 0, 1, 1, 0, 0, 0],
[ 0, 0, 0, 0, 0, 0, -1],
[ 0, 0, 0, 0, 1, -1, -1],
[ 0, 0, 0, 0, -1, 0, 0]
]);
C:=Matrix([
[-1, 0, 0, 0, 0, 0, 0],
[ 0, 1, 0, 0, 0, 0, 0],
[ 0, -1, -1, 0, 0, 0, 0],
[ 0, 0, 0, -1, 0, 0, 0],
[ 0, 0, 0, 0, -1, 0, 0],
[ 0, 0, 0, 0, 0, 1, 0],
[ 0, 0, 0, 0, 0, -1, -1]
]);
Order(A);
Order(B);
Order(C);
GL7Q:=GeneralLinearGroup(7,RationalField());
IdentifyGroup(sub<GL7Q | A,B,C>);
Automorphism(X,A);
Automorphism(X,B);
Automorphism(X,C);

// (48,41)
K<z_12>:=CyclotomicField(12);
i:=z_12^3;
c_9:=1;
P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
X:=Scheme(P6,[
x_0^2+ x_3*x_6 - x_4*x_5,
x_1^2 - x_2^2 + x_3*x_5 - x_4*x_6,
(c_9^2-2*i)*x_1*x_2 + i*(x_3*x_6 + x_4*x_5),
(c_9^2-2*i)*(x_1^2 + x_2^2) - c_9^2*(x_3*x_5 + x_4*x_6),
c_9*(x_1*x_3 + x_2*x_4)-i*x_5^2 - i*x_6^2,
c_9*(x_1*x_5 + x_2*x_6) +x_3^2 + x_4^2,
x_1*x_4 + x_2*x_3+c_9*x_5*x_6,
x_1*x_6 + x_2*x_5+i*c_9*(x_3*x_4),
(c_9^2-2*i)*(x_1*x_3 - x_2*x_4) +c_9*(i*x_5^2 - i*x_6^2),
(c_9^2-2*i)*(x_1*x_5 - x_2*x_6) -c_9*(x_3^2 - x_4^2)
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[-1, 0, 0, 0, 0, 0, 0],
[0, -1, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0],
[0, 0, 0, 0, 0, z_12^3, 0],
[0, 0, 0, 0, 0, 0, -z_12^3],
[0, 0, 0, -z_12^3, 0, 0, 0],
[0, 0, 0, 0, z_12^3, 0, 0]
]);
B:=Matrix([
[-1, 0, 0, 0, 0, 0, 0],
[0, 0, z_12^3, 0, 0, 0, 0],
[0, -z_12^3, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, z_12^2 - 1],
[0, 0, 0, 0, 0, -z_12^2 + 1, 0],
[0, 0, 0, 0, z_12^2, 0, 0],
[0, 0, 0, -z_12^2, 0, 0, 0]
]);
C:=Matrix([
[-1, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, z_12],
[0, 0, 0, 0, 0, z_12, 0],
[0, 0, 0, 0, -z_12^3 + z_12, 0, 0],
[0, 0, 0, -z_12^3 + z_12, 0, 0, 0]
]);
GL7K:=GeneralLinearGroup(7,K);
IdentifyGroup(sub<GL7K | A,B,C>);
Automorphism(X,A);
Automorphism(X,B);
Automorphism(X,C);

// (48,38) signature (2,2,2,4)
K<z_3>:=CyclotomicField(3);
P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
c_10:=1;
X:=Scheme(P6,[
x_0^2+ x_3*x_5 + x_4*x_6,
(2*c_10+1)*(x_1^2 + x_2^2) -c_10*(x_3*x_5 + x_4*x_6),
(2*c_10+1)*x_1*x_2 -c_10^2*(x_3*x_6 + x_4*x_5),
x_1^2 - x_2^2 + c_10*(x_3*x_5 - x_4*x_6),
(2*c_10+1)*(x_1*x_3 - x_2*x_4)-c_10*(x_5^2 - x_6^2),
(2*c_10+1)*(x_1*x_5 - x_2*x_6) -c_10*(x_3^2 - x_4^2),
x_1*x_3 + x_2*x_4 + c_10*(x_5^2 + x_6^2),
x_1*x_5 + x_2*x_6 + c_10*(x_3^2 + x_4^2),
x_1*x_4 + x_2*x_3 +x_5*x_6,
x_1*x_6 + x_2*x_5 + x_3*x_4
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[-1, 0, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0],
[0, 0, -1, 0, 0, 0, 0],
[0, 0, 0, 0, 0, z_3^2, 0],
[0, 0, 0, 0, 0, 0, -z_3^2],
[0, 0, 0, z_3, 0, 0, 0],
[0, 0, 0, 0, -z_3, 0, 0]
]);
B:=Matrix([
[-1, 0, 0, 0, 0, 0, 0],
[0, -1, 0, 0, 0, 0, 0],
[0, 0, -1, 0, 0, 0, 0],
[0, 0, 0, 0, 0, -1, 0],
[0, 0, 0, 0, 0, 0, -1],
[0, 0, 0, -1, 0, 0, 0],
[0, 0, 0, 0, -1, 0, 0]
]);
C:=Matrix([
[-1, 0, 0, 0, 0, 0, 0],
[0, 0, -1, 0, 0, 0, 0],
[0, -1, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, -1],
[0, 0, 0, 0, 0, -1, 0],
[0, 0, 0, 0, -1, 0, 0],
[0, 0, 0, -1, 0, 0, 0]
]);
Order(A);
Order(B);
Order(C);
GL7K:=GeneralLinearGroup(7,K);
IdentifyGroup(sub<GL7K | A,B,C>);
Automorphism(X,A);
Automorphism(X,B);


//(36,10) signature (2,2,2,6)
K:=RationalField();
c_3:=1;
c_6:=c_3+6;
c_8:=-1;
c_9:=-3;
c_11:=1;
P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
X:=Scheme(P6,[
x_0^2+c_3*(x_3^2 - x_3*x_4 - x_3*x_5 + 1/2*x_3*x_6 + x_4^2 + 1/2*x_4*x_5 + 1/2*x_4*x_6 + x_5^2 + 1/2*x_5*x_6 + x_6^2),
x_1^2 + x_1*x_2 + x_2^2+c_6*(x_3^2 - x_3*x_4 - x_3*x_5 + 1/2*x_3*x_6 + x_4^2 + 1/2*x_4*x_5 + 1/2*x_4*x_6 + x_5^2 + 1/2*x_5*x_6 + x_6^2),
x_0*x_1 + c_8*(x_1^2 + 2*x_1*x_2) + c_9*(-2*x_3*x_4 - x_3*x_6 + x_4^2 + x_4*x_5 - x_5*x_6 - x_6^2),
x_0*x_2 +c_8*(-2*x_1*x_2 - x_2^2)+c_9*(x_3^2 - x_3*x_5 + x_3*x_6 - x_4^2 - x_4*x_6 + x_5^2 + x_5*x_6),
x_1*x_3 + x_1*x_4 + 2*x_1*x_6 + 2*x_2*x_3 - x_2*x_4 + x_2*x_6 + c_11*(x_3^2 - x_3*x_4 + 2*x_3*x_5 + 2*x_3*x_6 + x_4^2 - x_4*x_5 + 2*x_4*x_6 - 2*x_5^2 - x_5*x_6 + x_6^2),
x_1*x_5 + 2*x_1*x_6 + 2*x_2*x_5 + x_2*x_6+ c_11*(2*x_3^2 -2*x_3*x_4 - 2*x_3*x_5 + x_3*x_6 + 2*x_4^2 + x_4*x_5 + x_4*x_6 - x_5^2- 2*x_5*x_6 - x_6^2),
x_0*x_3 + x_1*x_4 - x_1*x_5 - x_2*x_3 + x_2*x_4 - x_2*x_6 + x_3^2 + 2*x_3*x_4 + 2*x_3*x_5 + 2*x_3*x_6 - 2*x_4^2 - 4*x_4*x_5 - 4*x_4*x_6 - 2*x_5^2 - 4*x_5*x_6 - 2*x_6^2,
x_0*x_4 + x_1*x_3 - x_1*x_5 + x_2*x_4 + 2*x_3*x_4 - 2*x_3*x_6 - x_4^2 - 4*x_4*x_5 - 2*x_5*x_6 - 2*x_6^2,
x_0*x_5 -x_1*x_5 - x_2*x_5 - x_2*x_6 + 2*x_3^2 - 2*x_3*x_5 + 2*x_3*x_6 - 2*x_4^2 - 2*x_4*x_6 - x_5^2 - 4*x_5*x_6,
x_0*x_6 + x_1*x_5 + x_1*x_6 + x_2*x_6 + -4*x_3*x_4 - 2*x_3*x_6 + 2*x_4^2 + 2*x_4*x_5 + 4*x_5*x_6 + x_6^2
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[-1, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0],
[0, 0, 0, -1, -1, 0, 0],
[0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 0, 1, 0]
]);
B:=Matrix([
[-1, 0, 0, 0, 0, 0, 0],
[0, 1, -1, 0, 0, 0, 0],
[0, 0, -1, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 1, 0],
[0, 0, 0, 1, 1, 0, -1],
[0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 1, 0, 1, -1]
]);
C:=Matrix([
[-1, 0, 0, 0, 0, 0, 0],
[0, -1, 0, 0, 0, 0, 0],
[0, 0, -1, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 1, 0],
[0, 0, 0, -1, -1, -1, 1],
[0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 1]
]);
GL7K:=GeneralLinearGroup(7,K);
IdentifyGroup(sub<GL7K | A,B,C>);
Automorphism(X,A);
Automorphism(X,B);
Automorphism(X,C);



// (32,43) signature (2,2,2,8)
K<i>:=CyclotomicField(4);
P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
c_8:=1;
X:=Scheme(P6,[
x_0^2+ 2*x_3*x_4 - 2*i*x_5*x_6,
x_1^2 + x_2^2-2*x_3*x_4 +2*i*x_5*x_6,
x_3*x_6 - x_4*x_5,
x_1^2 - x_2^2+c_8*(x_3*x_4 + i*x_5*x_6),
x_0*x_1+x_3^2-x_4^2-i*x_5^2+i*x_6^2,
x_0*x_2-x_5^2-x_6^2-i*x_3^2-i*x_4^2,
x_0*x_3-x_1*x_4-i*x_2*x_4,
x_0*x_4+x_1*x_3-i*x_2*x_3,
x_0*x_5-i*x_2*x_6-x_1*x_6,
x_0*x_6-i*x_2*x_5+x_1*x_5
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[-1, 0, 0, 0, 0, 0, 0],
[0, -1, 0, 0, 0, 0, 0],
[0, 0, -1, 0, 0, 0, 0],
[0, 0, 0, -1, 0, 0, 0],
[0, 0, 0, 0, -1, 0, 0],
[0, 0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 0, 1]
]);
B:=Matrix([
[-1, 0, 0, 0, 0, 0, 0],
[0, 0, -1, 0, 0, 0, 0],
[0, -1, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, -i],
[0, 0, 0, 0, 0, -1, 0],
[0, 0, 0, 0, -1, 0, 0],
[0, 0, 0, i, 0, 0, 0]
]);
C:=Matrix([
[-1, 0, 0, 0, 0, 0, 0],
[0, -1, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0],
[0, 0, 0, 0, -i, 0, 0],
[0, 0, 0, i, 0, 0, 0],
[0, 0, 0, 0, 0, 0, -i],
[0, 0, 0, 0, 0, i, 0]
]);
D:=Matrix([
[-1, 0, 0, 0, 0, 0, 0],
[0, 0, -1, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, i, 0],
[0, 0, 0, 0, 0, 0, 1],
[0, 0, 0, -1, 0, 0, 0],
[0, 0, 0, 0, i, 0, 0]
]);
GL7K:=GeneralLinearGroup(7,K);
IdentifyGroup(sub<GL7K | A,B,C>);
Automorphism(X,A);
Automorphism(X,B);
Automorphism(X,C);


// (32,42)
K<z_8>:=CyclotomicField(8);
i:=z_8^2;
P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
c_8:=1;
c_10:=(1/2)*z_8^(-1)*c_8 -z_8; 
c_11:=(1/2)*z_8*c_8 -z_8^(-1);
X:=Scheme(P6,[
x_0^2+x_1^2 + x_2^2,
x_1*x_2+x_3*x_6 + i*x_4*x_5,
x_3*x_4+x_5*x_6,
x_1^2 - x_2^2+c_8*(x_3*x_6 - i*x_4*x_5),
x_0*x_1+c_10*(x_3^2 + x_4^2)+c_11*(x_5^2 -x_6^2),
x_0*x_2+c_10*(i*x_3^2 - i*x_4^2)+c_11*(i*x_5^2 + i*x_6^2),
x_0*x_5+z_8^3*(i*x_1*x_4 - x_2*x_4),
x_0*x_6+z_8^3*(x_1*x_3 -i*x_2*x_3),
x_0*x_3-z_8^3*(i*x_1*x_6 - x_2*x_6),
x_0*x_4-z_8^3*(x_1*x_5 -i*x_2*x_5)
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
GL7K:=GeneralLinearGroup(7,K);
A:=Matrix([
[-1,0,0,0,0,0,0],
[0,0,-1,0,0,0,0],
[0,-1,0,0,0,0,0],
[0,0,0,0,z_8,0,0],
[0,0,0,-z_8^3,0,0,0],
[0,0,0,0,0,0,z_8^3],
[0,0,0,0,0,-z_8,0]
]);
B:=Matrix([
[-1,0,0,0,0,0,0],
[0,1,0,0,0,0,0],
[0,0,-1,0,0,0,0],
[0,0,0,0,-z_8^2,0,0],
[0,0,0,z_8^2,0,0,0],
[0,0,0,0,0,0,-1],
[0,0,0,0,0,-1,0]
]);
C:=Matrix([
[-1,0,0,0,0,0,0],
[0,-1,0,0,0,0,0],
[0,0,-1,0,0,0,0],
[0,0,0,-1,0,0,0],
[0,0,0,0,1,0,0],
[0,0,0,0,0,1,0],
[0,0,0,0,0,0,-1]
]);
Order(A);
Order(B);
Order(C);
IdentifyGroup(sub<GL7K | A,B,C>);
Automorphism(X,A);
Automorphism(X,B);
Automorphism(X,C);