"Source: Text/Ring/FldAb.text";
"Line: 1943";
"Date: Fri Sep 26 12:10:11 2025";
"Main: Fri Sep 26 14:29:19 2025";
// original file: Text/Ring/FldAb.text, line: 1943
// Example: H41E10 ()
print "Example: H41E10";
ei := GetEchoInput();
SetEchoInput(true);
Zx<x> := PolynomialRing(Integers());
k := NumberField(x^2+1001);
g, m :=ClassGroup(k); g;
K := HilbertClassField(k);
o := MaximalOrder(k);
c := o`CyclotomicExtensions;
#c;
assert $1 eq 2;
c[1]`p2n; 
assert $1 eq 2;
c[2]`p2n;
assert $1 eq 5;
c[2]`Abs;
c[2]`Rel;
c[2]`Abs:Maximal;
Oa := c[2]`Abs;
z := c[2]`Zeta; z;
Oa!c[2]`Aut[1]`Abs(z);
z^c[2]`Aut[1]`r;
c[2]`Aut[1]`Order;
assert $1 eq 4;
 Oa!c[2]`Aut[1]`Abs(z);
 Oa!(c[2]`Aut[1]`Abs($1));
 Oa!(c[2]`Aut[1]`Abs($1));
 Oa!(c[2]`Aut[1]`Abs($1));
SetEchoInput(ei);
