\p 100

H=bnfinit(x^20 - 5*x^19 + 5*x^18 + 5*x^17 + 105*x^16 - 591*x^15 + 1545*x^14 - 1125*x^13 - 5975*x^12 + 28195*x^11 - 57199*x^10 + 44405*x^9 + 188910*x^8 - 778890*x^7 + 1946100*x^6 - 3335796*x^5 + 4553305*x^4 - 4695185*x^3 + 3627665*x^2 - 1817365*x + 443586);

\\ This gives the first class group.

print("The class groups of the degree 20 extension is ",H.cyc);

\\Now we take the subgroup of S5 to look at the fixed field.

P=x^5-85*x-153;
S=[[2,3,4,5,1]];
Q=subfieldgen(P,S);
L=bnfinit(Q);
Cl=bnrinit(L,idealmul(L,idealpow(L,idealprimedec(L,5)[1],5),idealpow(L,idealprimedec(L,5)[2],25)),1);
V=Test1(Cl,300);

print("The matrix of generators of the first test has size ", matsize(V));

print("This part should be run in different processes to speed up");

Cand=Test2(P,Cl,V,20000);

print("The characters having a Galois closure of degree at most 5^4 are:  ",Cand);

Cand2=Test3(P,Cl,V,Cand,200000);

print("The third test in each element gives the following number of order 30 elts:  ",Cand2);

print("This can we writen as the union of the spaces <[1,0,0,0,2],[0,0,0,1,3],[1,2,3,0,0]> <[1,0,4,0,3],[0,0,0,1,1]> and the first one has a 2-dimensional subspace which does not pass test 3")

print("The computation with a unique ramified prime gives");

Cl2=bnrinit(L,idealmul(L,idealpow(L,idealprimedec(L,5)[1],0),idealpow(L,idealprimedec(L,5)[2],22)),1);
V2=Test1(Cl2,300);

print("The matrix of generators of the first test has size ", matsize(V2));

Cand=Test2(P,Cl2,V2,20000);
print("The characters having a Galois closure of degree at most 5^4 are:  ",Cand);
Cand2=Test3(P,Cl2,V2,Cand,200000);
print("The third test in each element gives the following number of order 30 elts:  ",Cand2);

print("The evaluation of the first character at the primes above 113 is:");
vector(6,k,Cand[1]*V2~*Degree5Extensions(bnrisprincipal(Cl2,idealprimedec(L,113)[k],0),Cl2))~