Example 3.1:
[ Running Magma V2.21 ]

> K := QuadraticField(-21);
> ClassGroup(K);   
Abelian Group isomorphic to Z/2 + Z/2
Defined on 2 generators
Relations:
    2*$.1 = 0
    2*$.2 = 0
> f := HilbertClassPolynomial(-4*21*3^2);
> Temp := NumberField(f);               
> Temp := OptimizedRepresentation(Temp);
> f := MinimalPolynomial(Temp.1);
> H1 := ext < K  | f>;
> S := Subfields(H1);
> Kt := S[9][1]; // unique subfield of degree 3/K in this case
> R<x> := PolynomialRing(Integers());
> Kt := OptimizedRepresentation(NumberField(R ! MinimalPolynomial(K1.1)));
> R ! MinimalPolynomial(Kt.1);
x^3 - 6*x - 12
> // in paper


> f := WeberClassPolynomial(-4*21*3^4); // use smaller modular function 
> Temp := NumberField(f);
> Temp := Subfields(Temp,9)[1][1];
> Temp := OptimizedRepresentation(Temp);
> R ! MinimalPolynomial(Temp.1); // in paper
> OK := MaximalOrder(K);
> MinimalPolynomial(OK.2);
x^2 + 21
> alpha := 1+OK.2;
> P3 := Decomposition(OK,3)[1][1];
> Oloc,locmap := Completion(OK,P3);
> Oloc := ChangePrecision(Oloc,5);
> (Oloc ! locmap(alpha))^3; 
-8   
> // alpha^3 is congruent 1 mod 9, locally
> H := ext < K | MinimalPolynomial(Temp.1)>;
> H := AbelianExtension(H);
> A := ArtinMap(H);
> sigma := A(alpha);
> s := Automorphisms(NumberField(H)) ! sigma;
> K1 := FixedField(NumberField(H),[s]);
// find smaller polynomial
> Temp := NumberField(AbsoluteMinimalPolynomial(K1.1));
> Temp := Subfields(Temp,3)[1][1];
> Temp := OptimizedRepresentation(Temp);
> f := R ! MinimalPolynomial(Temp.1);
> assert IsIsomorphic(ext< K | f>, K1);
> f;
x^3 + 9*x - 12
> // in paper

-------------------------------------------------------

Example 3.2:
[ Running Magma V2.21 ]

> K := QuadraticField(-87);
> O := MaximalOrder(K);
> ClassGroup(O);
Abelian Group isomorphic to Z/6
Defined on 1 generator
Relations:
    6*$.1 = 0
> O1 := sub < O | 3>;
> PicardGroup(O1);
Abelian Group isomorphic to Z/18
Defined on 1 generator
Relations:
    18*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/18
Defined on 1 generator
Relations:
    18*$.1 = 0 to Set of ideals of O1 given by a rule [no inverse]
> f := WeberClassPolynomial(-87);
> Temp := NumberField(f);
> Temp := Subfields(Temp,3)[1][1];
> Temp := OptimizedRepresentation(Temp);
> R ! MinimalPolynomial(Temp.1);        
x^3 - x^2 + 2*x + 1
> // in paper

> f := WeberClassPolynomial(-87*3^2);
> Temp := NumberField(f);
> Temp := Subfields(Temp,9)[1][1];
> Temp := OptimizedRepresentation(Temp);
> R ! MinimalPolynomial(Temp.1);
x^9 + 3*x^8 + 6*x^7 + 14*x^6 + 9*x^5 + 21*x^4 + 6*x^3 + 12*x^2 + 3
> // in paper

> K := QuadraticField(-771);
> O := MaximalOrder(K);
> ClassGroup(O);
Abelian Group isomorphic to Z/6
Defined on 1 generator
Relations:
    6*$.1 = 0
> O1 := sub < O | 3>;
> PicardGroup(O1);
Abelian Group isomorphic to Z/3 + Z/6
Defined on 2 generators
Relations:
    3*$.1 = 0
    6*$.2 = 0
Mapping from: Abelian Group isomorphic to Z/3 + Z/6
Defined on 2 generators
Relations:
    3*$.1 = 0
    6*$.2 = 0 to Set of ideals of O1 given by a rule [no inverse]
> 

------------------------------------------

Example 3.3:
[ Running Magma V2.21 ]

> K := QuadraticField(-6789);
> O := MaximalOrder(K);
> O1 := sub < O | 3>;
> C,c := ClassGroup(O);

> TT := Subgroups(C);
> Sgrp := [];
> for k in TT do
for> if k`order eq 24 then
for|if> Sgrp[#Sgrp+1] := k;
for|if> end if;
for> end for;
> #Sgrp;
4
> S1 := Sgrp[1]`subgroup;
> S2 := Sgrp[2]`subgroup;
> S3 := Sgrp[3]`subgroup;
> S4 := Sgrp[4]`subgroup;
> P5 := Decomposition(O,5)[1][1]; 
> P7 := Decomposition(O,7)[1][1];  
> P11 := Decomposition(O,11)[1][1];
> P97 := Decomposition(O,97)[1][1];
> Inverse(c)(P5) in S2;
true
> Inverse(c)(P7) in S3;
true
> Inverse(c)(P11) in S4;
true
> Inverse(c)(P97) in S1;
true
> O1P5 := Decomposition(O1,5)[1][1];
> O1P7 := Decomposition(O1,7)[1][1]; 
> O1P11 := Decomposition(O1,11)[1][1];
> O1P97 := Decomposition(O1,97)[1][1];
> IsPrincipal(P11^6) and IsPrincipal(O1P11^6);
true
> P,p := PicardGroup(O1);
> I := p(6*P.3); 
> IsPrincipal(ideal < O | I>);
true
> // in paper

> for l in [1..3] do 
for> IsPrincipal(O1P5^(6*l));
for> end for;
false
false
true
> for l in [1..3] do
for> IsPrincipal((I*O1P5)^(6*l));
for> end for;
false
false
true
> for l in [1..3] do
for> IsPrincipal((I^2*O1P5)^(6*l));
for> end for;
false
false
true
> // same for O1P7, O1P97

> f := WeberClassPolynomial(-4*6789);
> Temp := NumberField(f);
> S := Subfields(Temp,3);
> #S;
4
> T1 := OptimizedRepresentation(S[1][1]);
> T2 := OptimizedRepresentation(S[2][1]);
> T3 := OptimizedRepresentation(S[3][1]);
> T4 := OptimizedRepresentation(S[4][1]);
> L1 := AbelianExtension(ext < K | MinimalPolynomial(T1.1)>);
> L2 := AbelianExtension(ext < K | MinimalPolynomial(T2.1)>);
> L3 := AbelianExtension(ext < K | MinimalPolynomial(T3.1)>);
> L4 := AbelianExtension(ext < K | MinimalPolynomial(T4.1)>);
> a1 := ArtinMap(L1);
> a2 := ArtinMap(L2);
> a3 := ArtinMap(L3);
> a4 := ArtinMap(L4);
> P11 := Decomposition(O,11)[1][1];
> NumberField(L1).1 eq a1(P11)(NumberField(L1).1);
false
> NumberField(L2).1 eq a2(P11)(NumberField(L2).1);
false
> NumberField(L3).1 eq a3(P11)(NumberField(L3).1);
true
> NumberField(L4).1 eq a4(P11)(NumberField(L4).1);
false
> R ! MinimalPolynomial(T3.1);
x^3 - x^2 + 8*x + 124
> // in paper

--------------------------------

Example 6.9:
[ Running Magma V2.21 ]

> L := CyclotomicField(9);
> zeta := L.1;
> R<x> := PolynomialRing(Integers());
> k2 := 3*((1-zeta^4)*(1-zeta^-4)/((1-zeta)*(1-zeta^-1)))^3;
> R ! MinimalPolynomial(k2);
x^3 - 1710*x^2 + 513*x - 27
> K := QuadraticField(-3);
> R<x> := PolynomialRing(K);
> f := x^27-1710*x^18+513*x^9-27;
> L := ext < K | f>;
> H := AbelianExtension(L);
> OK := MaximalOrder(K);
> L := NumberField(H);
> Generators(L);
{
    L.2,
    L.1
}
> P11 := Decomposition(OK,11)[1][1];
> f11 := FrobeniusAutomorphism(H,P11);
> f11(f11(f11(L.1)))-L.1;
0
> f11(f11(f11(L.2)))-L.2;
0
> eta := L.1+f11(L.1)+f11(f11(L.1));
> eta;
0
> eta := L.2+f11(L.2)+f11(f11(L.2));
> Degree(MinimalPolynomial(eta));
9
> f := MinimalPolynomial(eta);
x^9 - 59049*x^3 + 4251528*K.1
> // in paper
> // to verify:
> Temp := NumberField(f);
> R<x> := PolynomialRing(Temp);
> #Roots(R ! HilbertClassPolynomial(-3*3^6));
9

Example 6.10:
[ Running Magma V2.21 ]

> K := QuadraticField(-87);
> R<x> := PolynomialRing(Integers());
> f := x^2-727*x+1; // kappa_2 pol
> f1 := x^6-727*x^3+1;  
> L1 := NumberField(f1);
> f2 := x^2+3;
> L2 := NumberField(f2);
> L := CompositeFields(L1,L2)[1]; 
> for k in [1..#S] do; IsIsomorphic(S[1][1],S[k][1]); end for;
true Mapping from: Number Field with defining polynomial $.1^6 - 24*$.1^5 + 
    222*$.1^4 - 992*$.1^3 + 2193*$.1^2 - 2184*$.1 + 14270959 over the Rational
Field to Number Field with defining polynomial $.1^6 - 24*$.1^5 + 222*$.1^4 - 
    992*$.1^3 + 2193*$.1^2 - 2184*$.1 + 14270959 over the Rational Field
false
false
false
false
false
false
> // replacing S[1][1] by e.g. S[2][1] will yield more isomorphisms
> K1 := S[1][1];
> K1 := OptimizedRepresentation(M);
> R ! MinimalPolynomial(K1.1);
x^6 - 3*x^5 + 13*x^4 - 21*x^3 + 43*x^2 - 33*x + 9
> // in paper