"Source: Text/Ring/FldAb.text";
"Line: 1627";
"Date: Fri Sep 26 12:10:11 2025";
"Main: Fri Sep 26 14:29:19 2025";
// original file: Text/Ring/FldAb.text, line: 1627
// Example: H41E8 ()
print "Example: H41E8";
ei := GetEchoInput();
SetEchoInput(true);
// no-test
k := NumberField(Polynomial([-10, 0, 1]));
R, mR := RayClassGroup(4*3*5*MaximalOrder(k));
s := [x`subgroup : x in Subgroups(R:Quot := [2,2])];
a := [ AbelianExtension(Inverse(mq)*mR) where 
                            _, mq := quo<R|x> : x in s ];
n := [ x : x in a | IsNormal(x:All)];
ProbableAutomorphismGroup(n[2]);
g, c := ProbableAutomorphismGroup(n[1]);
#c;
assert $1 eq 2;
{* Order(x) : x in CosetImage(c[1], sub<c[1]|>) *};
{* Order(x) : x in CosetImage(c[2], sub<c[2]|>) *};
_ := ImproveAutomorphismGroup(n[1], c:Factor := 2);
SetEchoInput(ei);
