// (150,5) signature (2,3,10)
K<z_5>:=CyclotomicField(5);
P2<y_0,y_1,y_2>:=ProjectiveSpace(K,2);
X:=Scheme(P2,[y_0^5+y_1^5+y_2^5]);
IsSingular(X);
A:=Matrix([
[0,-z_5^2,0],
[-z_5^3,0,0],
[0,0,-1]
]);
B:=Matrix([
[0,0,z_5^2],
[1,0,0],
[0,z_5^3,0]
]);
Order(A);
Order(B);
Order( (A*B)^(-1));
GL3K:=GeneralLinearGroup(3,K);
IdentifyGroup(sub<GL3K | A,B>);
Automorphism(X,A);
Automorphism(X,B);


// (120,34) signature (2,4,6)
/* Inoue Kato equations
u_0:=-x_3-x_1-x_2;
u_1:=-x_4-x_2-x_0;
u_2:=-x_5-x_0-x_1;
u_3:=-x_0-x_4-x_5;
u_4:=-x_1-x_5-x_3;
u_5:=-x_2-x_3-x_4;
X:=Scheme(P5,[x_0*u_0-x_1*u_1,
x_0*u_0-x_2*u_2,
x_0*u_0+x_3*u_3,
x_0*u_0+x_4*u_4,
x_0*u_0+x_5*u_5,
x_0^2+u_0^2+x_1^2+u_1^2+x_2^2+u_2^2+x_3^2+u_3^2+x_4^2+u_4^2+x_5^2+u_5^2]);
*/
K:=RationalField();
P5<x_0,x_1,x_2,x_3,x_4,x_5>:=ProjectiveSpace(K,5);
X:=Scheme(P5,[
4*x_0^2 + 2*x_0*x_1 + 4*x_1^2 + 2*x_0*x_2 + 2*x_1*x_2 + 4*x_2^2 + 4*x_1*x_3 + 4*x_2*x_3 + 4*x_3^2 + 4*x_0*x_4 + 4*x_2*x_4 + 2*x_3*x_4 + 4*x_4^2 + 4*x_0*x_5 + 4*x_1*x_5 + 2*x_3*x_5 + 2*x_4*x_5 + 4*x_5^2,
-x_0*x_2 + x_1*x_2 - x_0*x_3 + x_1*x_4,
-x_0*x_1 + x_1*x_2 - x_0*x_3 + x_2*x_5,
-x_0*x_1 - x_0*x_2 - 2*x_0*x_3 - x_3*x_4 - x_3*x_5,
-x_0*x_1 - x_0*x_2 - x_0*x_3 - x_1*x_4 - x_3*x_4 - x_4*x_5,
-x_0*x_1 - x_0*x_2 - x_0*x_3 - x_2*x_5 - x_3*x_5 - x_4*x_5
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[0,0,0,0,-1,0],
[0,-1,1,0,-1,1],
[-1,0,0,0,0,1],
[-1,0,1,-1,0,1],
[-1,0,0,0,0,0],
[0,0,1,0,-1,0]
]);
B:=Matrix([
[0,0,1,0,-1,0],
[-1,0,1,-1,0,1],
[-1,0,0,0,0,0],
[-1,0,0,0,0,1],
[0,0,0,0,-1,0],
[0,-1,1,0,-1,1]
]);
Order(A);
Order(B);
Order( (A*B)^(-1));
GL6K:=GeneralLinearGroup(6,K);
IdentifyGroup(sub<GL6K | A,B>);
Automorphism(X,A);
Automorphism(X,B);

// (72,15) signature (2,4,9)
K<z_36>:=CyclotomicField(36);
z_12:=z_36^3;
z_9:=z_36^4;
z_6:=z_36^6;
z_3:=z_36^12;
P5<x_0,x_1,x_2,x_3,x_4,x_5>:=ProjectiveSpace(K,5);
X:=Scheme(P5,[
x_0*x_2-x_1^2,
x_0*x_4-x_1*x_3,
x_0*x_5-x_2*x_3,
x_1*x_4-x_0*x_5,
x_1*x_5-x_2*x_4,
x_3*x_5-x_4^2,
x_0^3+(4*z_6-2)*x_0^2*x_2+x_0*x_2^2+x_3^3+(-4*z_6+2)*x_3^2*x_5+x_3*x_5^2,
x_0^2*x_1+(4*z_6-2)*x_0*x_1*x_2+x_1*x_2^2+x_3^2*x_4+(-4*z_6+2)*x_3*x_4*x_5+x_4*x_5^2,
x_0^2*x_2+(4*z_6-2)*x_0*x_2^2+x_2^3+x_3^2*x_5+(-4*z_6+2)*x_3*x_5^2+x_5^3
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[0,  0,  0,  -z_9/2,  z_36^13/2,  z_9/2],
[0,  0,  0,  -z_36^13,  0,  -z_36^13],
[0,  0,  0,  z_9/2,  z_36^13/2,  -z_9/2],
[z_36^14/2,  z_36^5/2,  -z_36^14/2,  0,  0,  0],
[-z_36^5,  0,  -z_36^5,  0,  0,  0],
[-z_36^14/2,  z_36^5/2,  z_36^14/2,  0,  0,  0]
]);
B:=Matrix([
[0,  0,  0,  z_12^5,  0,  0],
[0,  0,  0,  0,  z_3^2,  0],
[0,  0,  0,  0,  0,  -z_12^5],
[z_12,  0,  0,  0,  0,  0],
[0,  z_3,  0,  0,  0,  0],
[0,  0,  -z_12,  0,  0,  0]
]);
Order(A);
Order(B);
Order( (A*B)^(-1));
GL6K:=GeneralLinearGroup(6,K);
IdentifyGroup(sub<GL6K | A,B>);
Automorphism(X,A);
Automorphism(X,B);

// (48,15) signature (2,6,8)
K<z_24>:=CyclotomicField(24);
z_8:=z_24^3;
i:=z_24^6;
z_6:=z_24^4;
z_3:=z_24^8;
P5<x_0,x_1,x_2,x_3,x_4,x_5>:=ProjectiveSpace(K,5);
X:=Scheme(P5,[
-x_4^2 + x_3*x_5,
-x_2*x_4 + x_1*x_5,
-x_2*x_3 + x_1*x_4,
-x_2*x_3 + x_0*x_5,
-x_1*x_3 + x_0*x_4,
-x_1^2 + x_0*x_2,
x_0*x_1^2 - x_2^3 - x_3*x_4^2 - x_5^3,
x_0^2*x_1 - x_1*x_2^2 - x_3^2*x_4 - x_4*x_5^2,
x_0^3 - x_1^2*x_2 - x_3^3 - x_4^2*x_5
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[0,0,0,0,0,-z_8],
[0,0,0,0,i,0],
[0,0,0,-z_8^3,0,0],
[0,0,z_8,0,0,0],
[0,-i,0,0,0,0],
[z_8^3,0,0,0,0,0]
]);
B:=Matrix([
[0,0,-z_6,0,0,0],
[0,z_6,0,0,0,0],
[-z_6,0,0,0,0,0],
[0,0,0,0,0,-z_3],
[0,0,0,0,z_3,0],
[0,0,0,-z_3,0,0]
]);
Order(A);
Order(B);
Order( (A*B)^(-1));
GL6K:=GeneralLinearGroup(6,K);
IdentifyGroup(sub<GL6K | A,B>);
Automorphism(X,A);
Automorphism(X,B);

// (39,1) signature (2,3,13)
K<z_13>:=CyclotomicField(13);
P2<y_0,y_1,y_2>:=ProjectiveSpace(K,2);
X:=Scheme(P2,[y_0^4*y_1+y_1^4*y_2+y_2^4*y_0]);
IsSingular(X);
A:=Matrix([
[0,0,z_13^12],
[z_13^4,0,0],
[0,z_13^10,0]
]);
B:=Matrix([
[0,z_13^7,0],
[0,0,z_13^11],
[z_13^8,0,0]
]);
Order(A);
Order(B);
Order( (A*B)^(-1));
GL3K:=GeneralLinearGroup(3,K);
IdentifyGroup(sub<GL3K | A,B>);
Automorphism(X,A);
Automorphism(X,B);


// (30,1) signature (2,10,15)
K<z_5>:=CyclotomicField(5);
P2<y_0,y_1,y_2>:=ProjectiveSpace(K,2);
X:=Scheme(P2,[y_0^5+y_1^4*y_2+2*z_5*y_1^3*y_2^2+2*z_5^2*y_1^2*y_2^3+z_5^3*y_1*y_2^4]);
IsSingular(X);
A:=Matrix([
[1,0,0],
[0,0,z_5^4],
[0,z_5,0]
]);
B:=Matrix([
[z_5^3,0,0],
[0,-z_5,0],
[0,-z_5^2,z_5]
]);
Order(A);
Order(B);
Order( (A*B)^(-1));
GL3K:=GeneralLinearGroup(3,K);
IdentifyGroup(sub<GL3K | A,B>);
Automorphism(X,A);
Automorphism(X,B);

// (21,2)
K<z_21>:=CyclotomicField(21);
z_7:=z_21^3;
z_3:=z_21^7;
P5<x_0,x_1,x_2,x_3,x_4,x_5>:=ProjectiveSpace(K,5);
X:=Scheme(P5,[
x_0*x_3-x_1*x_2,
x_0*x_4-x_1*x_3,
x_0*x_5-x_1*x_4,
x_2*x_4-x_3^2,
x_2*x_5-x_3*x_4,
x_3*x_5-x_4^2,
x_0^3-x_3*x_5^2 +x_2^3,
x_0^2*x_1-x_4*x_5^2 + x_2^2*x_3,
x_0*x_1^2-x_5^3+x_2^2*x_4
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=DiagonalMatrix([z_3^2,z_3^2,z_3,z_3,z_3,z_3]);
B:=DiagonalMatrix([1,z_7,1,z_7,z_7^2,z_7^3]);
GL6K:=GeneralLinearGroup(6,K);
Order(A);
Order(B);
Order( (A*B)^(-1));
IdentifyGroup(sub<GL6K | A,B>);
Automorphism(X,A);
Automorphism(X,B);



// (60,5) signature (2,2,2,3)
K<z_60>:=CyclotomicField(60);
p:=-z_60^14+z_60^6+z_60^4;
P5<x_0,x_1,x_2,x_3,x_4,x_5>:=ProjectiveSpace(K,5);
c_1:=1;
X:=Scheme(P5,[
x_0^2+x_0*x_1+p*x_0*x_2+x_1^2+(p-1)*x_1*x_2+x_2^2+c_1*(x_3^2-p*x_3*x_4-p*x_3*x_5+x_4^2+x_4*x_5+x_5^2),
24*x_0^2+24*x_0*x_3+(12*p+12)*x_0*x_5+(36*p+12)*x_1*x_4+(36*p+12)*x_1*x_5+(-8*p-16)*x_2^2+(-36*p-12)*x_2*x_4+(48*p+36)*x_2*x_5+(-288*p-180)*x_3^2+(396*p+252)*x_3*x_4+(540*p+324)*x_3*x_5+(-126*p-72)*x_4^2+(-252*p-144)*x_4*x_5+(-234*p-144)*x_5^2,
24*x_0*x_1+(-24*p-36)*x_0*x_3+(36*p+18)*x_0*x_4+(30*p+6)*x_0*x_5+(-54*p-36)*x_1*x_3+(24*p+18)*x_1*x_4+(42*p+18)*x_1*x_5+8*x_2^2+(-18*p-18)*x_2*x_3+(48*p+18)*x_2*x_4+(6*p+6)*x_2*x_5+(117*p+72)*x_3^2+(-108*p-72)*x_3*x_4+(-108*p-54)*x_3*x_5+(-18*p-9)*x_4^2+(-36*p-18)*x_4*x_5+(27*p+18)*x_5^2,
24*x_0*x_2+(6*p-6)*x_0*x_3+(-18*p-18)*x_0*x_4+12*p*x_0*x_5+(-6*p-12)*x_1*x_4+(-6*p-12)*x_1*x_5+(8*p+8)*x_2^2+18*p*x_2*x_3+(-12*p-6)*x_2*x_4+(-18*p-6)*x_2*x_5+(90*p+54)*x_3^2+(-18*p-18)*x_3*x_4+(-270*p-162)*x_3*x_5+(-45*p-27)*x_4^2+(72*p+54)*x_4*x_5+(153*p+99)*x_5^2,
(-24*p+12)*x_0*x_3+(36*p+12)*x_0*x_5+24*x_1^2+(36*p+36)*x_1*x_3+(-12*p-12)*x_1*x_4+(-12*p-12)*x_1*x_5+(8*p-24)*x_2^2+(-24*p+12)*x_2*x_4+(12*p+24)*x_2*x_5+(-216*p-126)*x_3^2+(360*p+216)*x_3*x_4+(216*p+108)*x_3*x_5+(-108*p-72)*x_4^2+(-216*p-144)*x_4*x_5+18*p*x_5^2,
(6*p+12)*x_0*x_3+(-24*p-18)*x_0*x_5+24*x_1*x_2+(18*p+18)*x_1*x_3+(-6*p+6)*x_1*x_4+(-24*p-12)*x_1*x_5+(8*p-16)*x_2^2+(-12*p-6)*x_2*x_4+(-18*p-6)*x_2*x_5+(-45*p-27)*x_3^2+(-18*p-18)*x_3*x_4+(108*p+54)*x_3*x_5+(90*p+54)*x_4^2+18*p*x_4*x_5+(-90*p-63)*x_5^2]
);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[-p, 0, 1, 0, 0, 0],
[0, -1, 0, 0, 0, 0],
[-p, 0, p, 0, 0, 0],
[0, 0, 0, -1, 0, 0],
[0, 0, 0, 0, 0, -1],
[0, 0, 0, 0, -1, 0]
]);
B:=Matrix([
[0, -1, 0, 0, 0, 0],
[-1, 0, 0, 0, 0, 0],
[1, -1, -1, 0, 0, 0],
[0, 0, 0, 0, 1, 0],
[0, 0, 0, 1, 0, 0],
[0, 0, 0, -p, -p, -1]
]);
C:=Matrix([
[-p, p-1, p-1, 0, 0, 0],
[-p, p-1, p, 0, 0, 0],
[-1, 1, 0, 0, 0, 0],
[0, 0, 0, -1, 0, -p],
[0, 0, 0, 0, -1, 1],
[0, 0, 0, 0, 0, 1]
]);
Order(A);
Order(B);
Order(C);
Order( (A*B*C)^(-1));
GL6K:=GeneralLinearGroup(6,K);
IdentifyGroup(sub<GL6K | A,B,C>);
Automorphism(X,A);
Automorphism(X,B);
Automorphism(X,C);

// (24,12) signature (2,2,3,4)
K<t>:=QuadraticField(3);
P5<x_0,x_1,x_2,x_3,x_4,x_5>:=ProjectiveSpace(K,5);
c_7:=2/t;
c_5:=t/2;
c_2:=13;
X:=Scheme(P5,[
x_0^2 - x_0*x_1 + x_0*x_2 + x_1^2 + x_2^2+c_2*(x_3^2 - x_3*x_5 + x_4^2 - x_4*x_5 + x_5^2),
x_0*x_3 + x_0*x_4 - 1/2*x_0*x_5 - 1/2*x_1*x_3 - 3/2*x_1*x_4 + x_1*x_5 + 3/2*x_2*x_3 + 1/2*x_2*x_4 - x_2*x_5+c_5*(x_3^2 - 2*x_3*x_4 + x_4^2 - x_5^2),
1/4*x_0*x_3 + 1/4*x_0*x_4 - 1/2*x_0*x_5 + 1/4*x_1*x_3 - 3/4*x_1*x_4 + 1/4*x_1*x_5 + 3/4*x_2*x_3 - 1/4*x_2*x_4 - 1/4*x_2*x_5+c_5*(1/2*x_3^2 + x_3*x_4 - x_3*x_5 + 1/2*x_4^2 - x_4*x_5),
x_0^2 + 2*x_0*x_2+ c_7*(x_0*x_3 - x_1*x_5) -2*x_4*x_5 + x_5^2,
x_0*x_1 + x_0*x_2 + c_7*(1/2*x_0*x_3 - 1/2*x_0*x_4 - 1/2*x_1*x_5 - 1/2*x_2*x_5) + x_3*x_5 - x_4*x_5,
x_1^2 - x_2^2 + c_7*(x_0*x_3 - x_0*x_4 + x_1*x_4 - x_1*x_5 + x_2*x_3 - x_2*x_5) + x_3^2 - x_4^2
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[-1, -1, 1, 0, 0, 0],
[0, 0, -1, 0, 0, 0],
[0, -1, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0],
[0, 0, 0, 1, 0, 0],
[0, 0, 0, -1, -1, -1]
]);
B:=Matrix([
[-1, 0, 1, 0, 0, 0],
[0, -1, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0],
[0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, -1, 0],
[0, 0, 0, 0, 1, 1]
]);
C:=Matrix([
[1, 1, 0, 0, 0, 0],
[-1, -1, 1, 0, 0, 0],
[1, 0, 0, 0, 0, 0],
[0, 0, 0, -1, -1, -1],
[0, 0, 0, 0, 0, -1],
[0, 0, 0, 1, 0, 1]
]);
Order(A);
Order(B);
Order(C);
Order( (A*B*C)^(-1));
GL6K:=GeneralLinearGroup(6,K);
IdentifyGroup(sub<GL6K | A,B,C>);
Automorphism(X,A);
Automorphism(X,B);
Automorphism(X,C);

// (24,8) signature (2,2,3,4)

K<z_12>:=CyclotomicField(12);
z_6:=z_12^2;
z_3:=z_12^4;
t:=17+z_12^5;
P5<x_0,x_1,x_2,x_3,x_4,x_5>:=ProjectiveSpace(K,5);
X:=Scheme(P5,[
x_0*x_2-x_1^2,
x_0*x_4-x_1*x_3,
x_0*x_5-x_2*x_3,
x_1*x_4-x_0*x_5,
x_1*x_5-x_2*x_4,
x_3*x_5-x_4^2,
x_0^3+t*x_0^2*x_2+x_0*x_2^2+x_3^3-t*x_3^2*x_5+x_3*x_5^2,
x_0^2*x_1+t*x_0*x_1*x_2+x_1*x_2^2+x_3^2*x_4-t*x_3*x_4*x_5+x_4*x_5^2,
x_0^2*x_2+t*x_0*x_2^2+x_2^3+x_3^2*x_5-t*x_3*x_5^2+x_5^3
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[0, 0, 0, 0, 0, z_12^5],
[0, 0, 0, 0, z_6, 0],
[0, 0, 0, z_12^-1, 0, 0],
[0, 0, z_12, 0, 0, 0],
[0, -z_6+1, 0, 0, 0, 0],
[-z_12, 0, 0, 0, 0, 0]
]);
B:=Matrix([
[0, 0, 1, 0, 0, 0],
[0, -1, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, -1, 0],
[0, 0, 0, 1, 0, 0]
]);
C:=Matrix([
[z_3, 0, 0, 0, 0, 0],
[0, z_3, 0, 0, 0, 0],
[0, 0, z_3, 0, 0, 0],
[0, 0, 0, z_3^2, 0, 0],
[0, 0, 0, 0, z_3^2, 0],
[0, 0, 0, 0, 0, z_3^2]
]);
Order(A);
Order(B);
Order(C);
Order( (A*B*C)^(-1));
GL6K:=GeneralLinearGroup(6,K);
IdentifyGroup(sub<GL6K | A,B,C>);
Automorphism(X,A);
Automorphism(X,B);
Automorphism(X,C);



// (24,6) signature (2,2,3,4) locus 17

K<z_24>:=CyclotomicField(24);
z_12:=z_24^2;
z_6:=z_24^4;
z_3:=z_6^2;
i:=z_24^6;
P5<x_0,x_1,x_2,x_3,x_4,x_5>:=ProjectiveSpace(K,5);
t:=17+z_24^5;
X:=Scheme(P5,[
x_0*x_2-x_1^2,
x_0*x_4-x_1*x_3,
x_1*x_4-x_2*x_3,
x_0*x_5-x_1*x_4,
x_1*x_5-x_2*x_4,
x_3*x_5-x_4^2,
x_0*x_2^2+t*x_0^3+t*x_5^2*x_3+x_3^3,
x_1*x_2^2+t*x_0^2*x_1+t*x_5^2*x_4+x_3^2*x_4,
x_2^3+t*x_0^2*x_2+t*x_5^3+x_3^2*x_5
]);
Dimension(X);
IsSingular(X);
HilbertPolynomial(Ideal(X));
A:=Matrix([
[0,0,0,0,0,-z_12^-1],
[0,0,0,0,z_6,0],
[0,0,0,z_12^-1,0,0],
[0,0,z_12,0,0,0],
[0,-z_3,0,0,0,0],
[-z_12,0,0,0,0,0]
]);
B:=Matrix([
[0,0,0,0,0,-z_3],
[0,0,0,0,z_3,0],
[0,0,0,-z_3,0,0],
[0,0,z_6,0,0,0],
[0,-z_6,0,0,0,0],
[z_6,0,0,0,0,0]
]);
C:=DiagonalMatrix([z_3^2,z_3^2,z_3^2,z_3,z_3,z_3]);
Order(A);
Order(B);
Order(C);
Order( (A*B*C)^-1);
GL6K:=GeneralLinearGroup(6,K);
IdentifyGroup(sub<GL6K | A,B,C>);
Automorphism(X,A);
Automorphism(X,B);
Automorphism(X,C);