SetClassGroupBounds("GRH");

n:=16;
r:=(n div 2);  // r:=n div 2;  if cyclotomic
R<x> := PolynomialRing(Integers());
f:=x^n+1*x^ 0 ;
K<x> := NumberField(f);
d:=Discriminant(K);
d;
Abs(d)^(1/n);

o:=EquationOrder(K);
O:=MaximalOrder(o);
TransformationMatrix(O,o);
C,mC:=ClassGroup(O:Proof:="GRH");
C;

mC(C.1);

U,mU:=UnitGroup(O: GRH:=true);

for i in [1..r] do
 Ui:=mU(U.i);Ui;
end for;

Regulator(O:Current);




SetClassGroupBounds("GRH");

n:=16;
r:=(n div 2)+1;  // r:=n div 2;  if cyclotomic
R<x> := PolynomialRing(Integers());
f:=x^n-2*x^ 0 ;
K<x> := NumberField(f);
d:=Discriminant(K);
d;
Abs(d)^(1/n);

o:=EquationOrder(K);
O:=MaximalOrder(o);
TransformationMatrix(O,o);
C,mC:=ClassGroup(O:Proof:="GRH");
C;

mC(C.1);

U,mU:=UnitGroup(O: GRH:=true);

for i in [1..r] do
 Ui:=mU(U.i);Ui;
end for;

Regulator(O:Current);








SetClassGroupBounds("GRH");

n:=16;
R<x> := PolynomialRing(Integers());
f:=x^n-229*x^ 0 ;
K<x> := NumberField(f);
d:=Discriminant(K);
d;
Abs(d)^(1/n);

o:=EquationOrder(K);
O:=MaximalOrder(o);
TransformationMatrix(O,o);
C,mC:=ClassGroup(O:Proof:="GRH");
C;

mC(C.1);

U,mU:=UnitGroup(O: GRH:=true);

U1:=mU(U.1);U1;
U2:=mU(U.2);U2;
U3:=mU(U.3);U3;
U4:=mU(U.4);U4;
U5:=mU(U.5);U5;
U6:=mU(U.6);U6;
U7:=mU(U.7);U7;
U8:=mU(U.8);U8;


I:=2*O;
F:=Factorization(I);F;

I1:=F[1][1]^5;
I1;

V,om:=IsPrincipal(I1);



om;
Factorization((om-2^5)*O+F[1][1]^7)[1][2],Factorization((om+2^5)*O+F[1][1]^7)[1][2];
Factorization((om-1)*O+F[2][1]^2)[1][2],Factorization((om-3)*O+F[2][1]^2)[1][2];
Factorization((om-1)*O+F[3][1]^2)[1][2],Factorization((om-3)*O+F[3][1]^2)[1][2];
Factorization((om-1)*O+F[4][1]^2)[1][2],Factorization((om-3)*O+F[4][1]^2)[1][2];
Factorization((om-1)*O+F[5][1]^2)[1][2],Factorization((om-3)*O+F[5][1]^2)[1][2];
Factorization((om-1)*O+F[6][1]^2)[1][2],Factorization((om-3)*O+F[6][1]^2)[1][2];
Factorization((om-1)*O+F[7][1]^2)[1][2],Factorization((om-3)*O+F[7][1]^2)[1][2];
Factorization((om-1)*O+F[8][1]^2)[1][2],Factorization((om-3)*O+F[8][1]^2)[1][2];

U1;
Factorization((U1-1)*O+F[1][1]^2)[1][2],Factorization((U1-3)*O+F[1][1]^2)[1][2];
Factorization((U1-1)*O+F[2][1]^2)[1][2],Factorization((U1-3)*O+F[2][1]^2)[1][2];
Factorization((U1-1)*O+F[3][1]^2)[1][2],Factorization((U1-3)*O+F[3][1]^2)[1][2];
Factorization((U1-1)*O+F[4][1]^2)[1][2],Factorization((U1-3)*O+F[4][1]^2)[1][2];
Factorization((U1-1)*O+F[5][1]^2)[1][2],Factorization((U1-3)*O+F[5][1]^2)[1][2];
Factorization((U1-1)*O+F[6][1]^2)[1][2],Factorization((U1-3)*O+F[6][1]^2)[1][2];
Factorization((U1-1)*O+F[7][1]^2)[1][2],Factorization((U1-3)*O+F[7][1]^2)[1][2];
Factorization((U1-1)*O+F[8][1]^2)[1][2],Factorization((U1-3)*O+F[8][1]^2)[1][2];

U2;
Factorization((U2-1)*O+F[1][1]^2)[1][2],Factorization((U2-3)*O+F[1][1]^2)[1][2];
Factorization((U2-1)*O+F[2][1]^2)[1][2],Factorization((U2-3)*O+F[2][1]^2)[1][2];
Factorization((U2-1)*O+F[3][1]^2)[1][2],Factorization((U2-3)*O+F[3][1]^2)[1][2];
Factorization((U2-1)*O+F[4][1]^2)[1][2],Factorization((U2-3)*O+F[4][1]^2)[1][2];
Factorization((U2-1)*O+F[5][1]^2)[1][2],Factorization((U2-3)*O+F[5][1]^2)[1][2];
Factorization((U2-1)*O+F[6][1]^2)[1][2],Factorization((U2-3)*O+F[6][1]^2)[1][2];
Factorization((U2-1)*O+F[7][1]^2)[1][2],Factorization((U2-3)*O+F[7][1]^2)[1][2];
Factorization((U2-1)*O+F[8][1]^2)[1][2],Factorization((U2-3)*O+F[8][1]^2)[1][2];

U3;
Factorization((U3-1)*O+F[1][1]^2)[1][2],Factorization((U3-3)*O+F[1][1]^2)[1][2];
Factorization((U3-1)*O+F[2][1]^2)[1][2],Factorization((U3-3)*O+F[2][1]^2)[1][2];
Factorization((U3-1)*O+F[3][1]^2)[1][2],Factorization((U3-3)*O+F[3][1]^2)[1][2];
Factorization((U3-1)*O+F[4][1]^2)[1][2],Factorization((U3-3)*O+F[4][1]^2)[1][2];
Factorization((U3-1)*O+F[5][1]^2)[1][2],Factorization((U3-3)*O+F[5][1]^2)[1][2];
Factorization((U3-1)*O+F[6][1]^2)[1][2],Factorization((U3-3)*O+F[6][1]^2)[1][2];
Factorization((U3-1)*O+F[7][1]^2)[1][2],Factorization((U3-3)*O+F[7][1]^2)[1][2];
Factorization((U3-1)*O+F[8][1]^2)[1][2],Factorization((U3-3)*O+F[8][1]^2)[1][2];

U4;
Factorization((U4-1)*O+F[1][1]^2)[1][2],Factorization((U4-3)*O+F[1][1]^2)[1][2];
Factorization((U4-1)*O+F[2][1]^2)[1][2],Factorization((U4-3)*O+F[2][1]^2)[1][2];
Factorization((U4-1)*O+F[3][1]^2)[1][2],Factorization((U4-3)*O+F[3][1]^2)[1][2];
Factorization((U4-1)*O+F[4][1]^2)[1][2],Factorization((U4-3)*O+F[4][1]^2)[1][2];
Factorization((U4-1)*O+F[5][1]^2)[1][2],Factorization((U4-3)*O+F[5][1]^2)[1][2];
Factorization((U4-1)*O+F[6][1]^2)[1][2],Factorization((U4-3)*O+F[6][1]^2)[1][2];
Factorization((U4-1)*O+F[7][1]^2)[1][2],Factorization((U4-3)*O+F[7][1]^2)[1][2];
Factorization((U4-1)*O+F[8][1]^2)[1][2],Factorization((U4-3)*O+F[8][1]^2)[1][2];

U5;
Factorization((U5-1)*O+F[1][1]^2)[1][2],Factorization((U5-3)*O+F[1][1]^2)[1][2];
Factorization((U5-1)*O+F[2][1]^2)[1][2],Factorization((U5-3)*O+F[2][1]^2)[1][2];
Factorization((U5-1)*O+F[3][1]^2)[1][2],Factorization((U5-3)*O+F[3][1]^2)[1][2];
Factorization((U5-1)*O+F[4][1]^2)[1][2],Factorization((U5-3)*O+F[4][1]^2)[1][2];
Factorization((U5-1)*O+F[5][1]^2)[1][2],Factorization((U5-3)*O+F[5][1]^2)[1][2];
Factorization((U5-1)*O+F[6][1]^2)[1][2],Factorization((U5-3)*O+F[6][1]^2)[1][2];
Factorization((U5-1)*O+F[7][1]^2)[1][2],Factorization((U5-3)*O+F[7][1]^2)[1][2];
Factorization((U5-1)*O+F[8][1]^2)[1][2],Factorization((U5-3)*O+F[8][1]^2)[1][2];

U6;
Factorization((U6-1)*O+F[1][1]^2)[1][2],Factorization((U6-3)*O+F[1][1]^2)[1][2];
Factorization((U6-1)*O+F[2][1]^2)[1][2],Factorization((U6-3)*O+F[2][1]^2)[1][2];
Factorization((U6-1)*O+F[3][1]^2)[1][2],Factorization((U6-3)*O+F[3][1]^2)[1][2];
Factorization((U6-1)*O+F[4][1]^2)[1][2],Factorization((U6-3)*O+F[4][1]^2)[1][2];
Factorization((U6-1)*O+F[5][1]^2)[1][2],Factorization((U6-3)*O+F[5][1]^2)[1][2];
Factorization((U6-1)*O+F[6][1]^2)[1][2],Factorization((U6-3)*O+F[6][1]^2)[1][2];
Factorization((U6-1)*O+F[7][1]^2)[1][2],Factorization((U6-3)*O+F[7][1]^2)[1][2];
Factorization((U6-1)*O+F[8][1]^2)[1][2],Factorization((U6-3)*O+F[8][1]^2)[1][2];

U7;
Factorization((U7-1)*O+F[1][1]^2)[1][2],Factorization((U7-3)*O+F[1][1]^2)[1][2];
Factorization((U7-1)*O+F[2][1]^2)[1][2],Factorization((U7-3)*O+F[2][1]^2)[1][2];
Factorization((U7-1)*O+F[3][1]^2)[1][2],Factorization((U7-3)*O+F[3][1]^2)[1][2];
Factorization((U7-1)*O+F[4][1]^2)[1][2],Factorization((U7-3)*O+F[4][1]^2)[1][2];
Factorization((U7-1)*O+F[5][1]^2)[1][2],Factorization((U7-3)*O+F[5][1]^2)[1][2];
Factorization((U7-1)*O+F[6][1]^2)[1][2],Factorization((U7-3)*O+F[6][1]^2)[1][2];
Factorization((U7-1)*O+F[7][1]^2)[1][2],Factorization((U7-3)*O+F[7][1]^2)[1][2];
Factorization((U7-1)*O+F[8][1]^2)[1][2],Factorization((U7-3)*O+F[8][1]^2)[1][2];


U8;
Factorization((U8-1)*O+F[1][1]^2)[1][2],Factorization((U8-3)*O+F[1][1]^2)[1][2];
Factorization((U8-1)*O+F[2][1]^2)[1][2],Factorization((U8-3)*O+F[2][1]^2)[1][2];
Factorization((U8-1)*O+F[3][1]^2)[1][2],Factorization((U8-3)*O+F[3][1]^2)[1][2];
Factorization((U8-1)*O+F[4][1]^2)[1][2],Factorization((U8-3)*O+F[4][1]^2)[1][2];
Factorization((U8-1)*O+F[5][1]^2)[1][2],Factorization((U8-3)*O+F[5][1]^2)[1][2];
Factorization((U8-1)*O+F[6][1]^2)[1][2],Factorization((U8-3)*O+F[6][1]^2)[1][2];
Factorization((U8-1)*O+F[7][1]^2)[1][2],Factorization((U8-3)*O+F[7][1]^2)[1][2];
Factorization((U8-1)*O+F[8][1]^2)[1][2],Factorization((U8-3)*O+F[8][1]^2)[1][2];




M1:=Transpose(Matrix(GF(2), 8, 8, 
[
1,1,1,1,1,1,1,1,
1,1,0,1,0,0,1,0,
1,1,0,0,0,1,0,1,
1,1,0,0,1,0,0,1,
0,0,1,0,1,0,1,1,
1,0,1,0,0,1,1,0,
1,0,1,0,0,1,0,1,
0,1,1,1,0,1,0,1
]));
M1;

v1:=Transpose(Matrix(GF(2), 1, 8, [0,0,0,1,0,0,0,0]));
v1;

M1^-1*v1;

omega:=om*U1*U2*U3*U5*U6*U8;


omega;
Factorization((omega-2^5)*O+F[1][1]^7)[1][2],Factorization((omega+2^5)*O+F[1][1]^7)[1][2];
Factorization((omega-1)*O+F[2][1]^2)[1][2],Factorization((omega-3)*O+F[2][1]^2)[1][2];
Factorization((omega-1)*O+F[3][1]^2)[1][2],Factorization((omega-3)*O+F[3][1]^2)[1][2];
Factorization((omega-1)*O+F[4][1]^2)[1][2],Factorization((omega-3)*O+F[4][1]^2)[1][2];
Factorization((omega-1)*O+F[5][1]^2)[1][2],Factorization((omega-3)*O+F[5][1]^2)[1][2];
Factorization((omega-1)*O+F[6][1]^2)[1][2],Factorization((omega-3)*O+F[6][1]^2)[1][2];
Factorization((omega-1)*O+F[7][1]^2)[1][2],Factorization((omega-3)*O+F[7][1]^2)[1][2];
Factorization((omega-1)*O+F[8][1]^2)[1][2],Factorization((omega-3)*O+F[8][1]^2)[1][2];



A:=Automorphisms(K);
for phi in A do phi(omega); end for;


phi:=A[2];
phi(x);
(phi^2)(x);
(phi^4)(x);
(phi^8)(x);


omega0:=omega;
omega1:=omega*phi(omega);
omega2:=omega*((phi^2)(omega));

omega2;
Conjugate(omega2,1);
Conjugate(omega2,2);
Conjugate(omega2,3);
Conjugate(omega2,4);
Conjugate(omega2,5);
Conjugate(omega2,6);
Conjugate(omega2,7);
Conjugate(omega2,8);

omega1;
Conjugate(omega1,1);
Conjugate(omega1,2);
Conjugate(omega1,3);
Conjugate(omega1,4);
Conjugate(omega1,5);
Conjugate(omega1,6);
Conjugate(omega1,7);
Conjugate(omega1,8);

omega0;
Conjugate(omega0,1);
Conjugate(omega0,2);
Conjugate(omega0,3);
Conjugate(omega0,4);
Conjugate(omega0,5);
Conjugate(omega0,6);
Conjugate(omega0,7);
Conjugate(omega0,8);



omega;
Factorization((omega-2^5)*O+F[1][1]^7)[1][2];
Factorization((omega+2^5)*O+F[1][1]^7)[1][2];
Factorization((omega-1)*O+F[2][1]^2)[1][2];
Factorization((omega-3)*O+F[2][1]^2)[1][2];
Factorization((omega-1)*O+F[3][1]^2)[1][2];
Factorization((omega-3)*O+F[3][1]^2)[1][2];
Factorization((omega-1)*O+F[4][1]^2)[1][2];
Factorization((omega-3)*O+F[4][1]^2)[1][2];
Factorization((omega-1)*O+F[5][1]^2)[1][2];
Factorization((omega-3)*O+F[5][1]^2)[1][2];
Factorization((omega-1)*O+F[6][1]^2)[1][2];
Factorization((omega-3)*O+F[6][1]^2)[1][2];
Factorization((omega-1)*O+F[7][1]^2)[1][2];
Factorization((omega-3)*O+F[7][1]^2)[1][2];
Factorization((omega-1)*O+F[8][1]^2)[1][2];
Factorization((omega-3)*O+F[8][1]^2)[1][2];