// (120,34)
K:=RationalField();
P3<x_0,x_1,x_2,x_3>:=ProjectiveSpace(K,3);
X:=Scheme(P3,[x_0^2 + x_0*x_1 + x_1^2 - x_1*x_2 + x_2^2 - x_2*x_3 + x_3^2,
x_0^2*x_1 + x_0*x_1^2 + x_1^2*x_2 - x_1*x_2^2 + x_2^2*x_3 - x_2*x_3^2]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[-1,0,0,0],
[0,-1,0,0],
[0,0,-1,-1],
[0,0,0,1]
]);
B:=Matrix([
[1,-1,-1,0],
[1,0,0,0],
[0,-1,0,0],
[0,0,-1,-1]
]);
Order(A);
Order(B);
Order((A*B)^-1);
GL4K:=GeneralLinearGroup(4,K);
IdentifyGroup(sub<GL4K | A,B>);
Automorphism(X,A);
Automorphism(X,B);

R=QQ[x_0..x_3];
I=ideal({
x_0^2 + x_0*x_1 + x_1^2 - x_1*x_2 + x_2^2 - x_2*x_3 + x_3^2,
x_0^2*x_1 + x_0*x_1^2 + x_1^2*x_2 - x_1*x_2^2 + x_2^2*x_3 - x_2*x_3^2
});
A=map(R,R,{-x_0,-x_1,-x_2,-x_2+x_3})
B=map(R,R,{x_0+x_1,-x_0-x_2,-x_0-x_3,-x_3})

// (72,42)
K<z_12>:=CyclotomicField(12);
z_6:=z_12^2;
z_3:=z_12^4;
i:=z_12^3;
P3<x_0,x_1,x_2,x_3>:=ProjectiveSpace(K,3);
X:=Scheme(P3,[
-x_2^2 + x_1*x_3,
x_0^3 - x_1^2*x_2 +x_2*x_3^2
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[-1,0,0,0],
[0,0,0,-i],
[0,0,-1,0],
[0,i,0,0]
]);
B:=Matrix([
[-z_6,0,0,0],
[0,-1/2*z_12,1/2*z_3,1/2*z_12],
[0,z_12,0,z_12],
[0,-1/2*z_12,-1/2*z_3,1/2*z_12]
]);
Order(A);
Order(B);
Order( (A*B)^-1);
GL4K:=GeneralLinearGroup(4,K);
IdentifyGroup(sub<GL4K | A,B>);
Automorphism(X,A);
Automorphism(X,B);

loadPackage("Cyclotomic");
K=cyclotomicField(24);
z_24=K_0;
z_12=z_24^2;
z_6=z_24^4;
i=z_24^6;
z_3=z_24^8;
R=K[x_0..x_3];
I=ideal({-x_2^2 + x_1*x_3,
x_0^3 - x_1^2*x_2 +x_2*x_3^2});
A=map(R,R,{-x_0,i*x_3,-x_2,-i*x_1});
B=map(R,R,{-z_6*x_0,-1/2*z_12*x_1+z_12*x_2-1/2*z_12*x_3,1/2*z_3*x_1-1/2*z_3*x_3,1/2*z_12*x_1+z_12*x_2+1/2*z_12*x_3})
B=map(R,R,{-z_6*x_0,-1/2*z_12*(x_1-2*x_2+x_3),1/2*z_3*(x_1-x_3),1/2*z_12*(x_1+2*x_2+x_3)})


// (72,40)
K<z_6>:=CyclotomicField(6);
P3<x_0,x_1,x_2,x_3>:=ProjectiveSpace(K,3);
X:=Scheme(P3,[
x_0*x_3-x_1*x_2,
x_1^3-x_0^3-x_3^3-x_2^3]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[-1,0,0,0],
[0,0,1,0],
[0,1,0,0],
[0,0,0,-1]
]);
B:=Matrix([
[0,z_6^2,0,0],
[0,0,0,1],
[-1,0,0,0],
[0,0,z_6,0]
]);
GL4K:=GeneralLinearGroup(4,K);
IdentifyGroup(sub<GL4K | A,B>);
Automorphism(X,A);
Automorphism(X,B);


loadPackage("Cyclotomic");
K=cyclotomicField(6);
z_6=K_0;
R=K[x_0..x_3];
I=ideal({x_0*x_3-x_1*x_2,
x_1^3-x_0^3-x_3^3-x_2^3});
A=map(R,R,{-x_0,x_2,x_1,-x_3});
B=map(R,R,{-x_2,z_6^2*x_0,z_6*x_3,x_1});
A(I)==I
B(I)==I

// (36,12)
K<z_6>:=CyclotomicField(6);
z_3:=z_6^2;
P3<x_0,x_1,x_2,x_3>:=ProjectiveSpace(K,3);
X:=Scheme(P3,[
x_1*x_3-x_2^2,
x_0^3-x_1^3+x_3^3]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[-1,0,0,0],
[0,0,0,-z_6],
[0,0,-1,0],
[0,z_6-1,0,0]
]);
B:=Matrix([
[z_3,0,0,0],
[0,0,0,-1],
[0,0,z_6,0],
[0,-z_3,0,0]
]);
Order(A);
Order(B);
Order( (A*B)^-1);
GL4K:=GeneralLinearGroup(4,K);
IdentifyGroup(sub<GL4K | A,B>);
Automorphism(X,A);
Automorphism(X,B);


loadPackage("Cyclotomic");
K=cyclotomicField(6);
z_6=K_0;
z_3=z_6^2;
R=K[x_0..x_3];
I=ideal({x_1*x_3-x_2^2,
x_0^3-x_1^3+x_3^3});
A=map(R,R,{-x_0,z_3*x_3,-x_2,-z_6*x_1});
B=map(R,R,{z_3*x_0,-z_3*x_3, z_6*x_2, -x_1});
A(I)==I
B(I)==I


// (15,1)
K<z_15>:=CyclotomicField(15);
P3<x_0,x_1,x_2,x_3>:=ProjectiveSpace(K,3);
X:=Scheme(P3,[
-x_2^2 + x_1*x_3,
x_0^3 - x_1^2*x_2 +x_3^3
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=DiagonalMatrix([z_15^10,z_15^5,z_15^5,z_15^5]);
B:=DiagonalMatrix([z_15^3,z_15^9,z_15^6,z_15^3]);
Order(A);
Order(B);
Order( (A*B)^-1);
GL4K:=GeneralLinearGroup(4,K);
IdentifyGroup(sub<GL4K | A,B>);
Automorphism(X,A);
Automorphism(X,B);



loadPackage("Cyclotomic");
K=cyclotomicField(15);
z_15=K_0;
z_5=z_15^3;
z_3=z_15^5;
R=K[x_0..x_3];
I=ideal({x_1*x_3-x_2^2,
x_0^3 - x_1^2*x_2 +x_3^3});
A=map(R,R,{z_3^2*x_0,z_3* x_1,z_3*x_2, z_3*x_3});
B=map(R,R,{z_5*x_0,z_5^3*x_1, z_5^2*x_2, z_5*x_3});
A(I)==I
B(I)==I




// (36,10)
K<z_3>:=CyclotomicField(3);
P3<x_0,x_1,x_2,x_3>:=ProjectiveSpace(K,3);
t:=17;
X:=Scheme(P3,[
x_0*x_3-x_1*x_2,
x_1^3 + t*x_0^3 + x_2^3 - t*x_3^3
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[-1,0,0,0],
[0,0,-z_3,0],
[0,-z_3^2,0,0],
[0,0,0,-1]
]);
B:=Matrix([
[0,0,0,1],
[0,-1,0,0],
[0,0,-1,0],
[1,0,0,0]
]);
C:=Matrix([
[0,0,0,-z_3^2],
[0,0,1,0],
[0,1,0,0],
[-z_3,0,0,0]
]);
Order(A);
Order(B);
Order(C);
Order( (A*B*C)^-1);
GL4K:=GeneralLinearGroup(4,K);
IdentifyGroup(sub<GL4K | A,B,C>);
Automorphism(X,A);
Automorphism(X,B);
Automorphism(X,C);

// (24,12)
K:=RationalField();
P3<x_0,x_1,x_2,x_3>:=ProjectiveSpace(K,3);
t:=17;
X:=Scheme(P3,[
x_0^2+x_1^2+x_2^2+t*x_3^2,
x_0*x_1*x_2-x_3^3
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[-1,0,0,0],
[0,0,-1,0],
[0,-1,0,0],
[0,0,0,-1]
]);
B:=Matrix([
[0,0,1,0],
[0,-1,0,0],
[1,0,0,0],
[0,0,0,-1]
]);
C:=Matrix([
[-1,0,0,0],
[0,0,1,0],
[0,1,0,0],
[0,0,0,-1]
]);
Order(A);
Order(B);
Order(C);
Order( (A*B*C)^-1);
GL4K:=GeneralLinearGroup(4,K);
IdentifyGroup(sub<GL4K | A,B,C>);
Automorphism(X,A);
Automorphism(X,B);
Automorphism(X,C);


// (18,3)
K<z_3>:=CyclotomicField(3);
P3<x_0,x_1,x_2,x_3>:=ProjectiveSpace(K,3);
t:=17+z_3;
X:=Scheme(P3,[x_1*x_3-x_2^2,x_0^3-x_3^3+t*x_2^3+x_1^3]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[-1,0,0,0],
[0,0,0,1],
[0,0,-1,0],
[0,1,0,0]
]);
B:=Matrix([
[-1,0,0,0],
[0,0,0,z_3^2],
[0,0,-1,0],
[0,z_3,0,0]
]);
C:=DiagonalMatrix([1,z_3^2,z_3,1]);
Order(A);
Order(B);
Order(C);
Order( (A*B*C)^-1);
GL4K:=GeneralLinearGroup(4,K);
IdentifyGroup(sub<GL4K | A,B,C>);
Automorphism(X,A);
Automorphism(X,B);
Automorphism(X,C);