P1 = x^4-5;
polgalois(P1)

P2 = x^4-x^3-7*x^2+2*x+9;
polgalois(P2)

P3 = x^4-x^3-3*x^2+x-1;
polgalois(P3)

Q1 = nfsplitting(P1)
Q2 = nfsplitting(P2)
  + 51997*x^4 - 79707*x^2 + 26569

Q3 = nfsplitting(P3)
 +54012*x^19+51764*x^18-1348092*x^17-2201841*x^16
 +21708244*x^15+41344014*x^14-241723272*x^13
 -454688929*x^12+1972336584*x^11+3130578366*x^10
 -12348327032*x^9-13356023346*x^8+59757161004*x^7
 +32173517686*x^6-204540935496*x^5-11176476888*x^4
 +433089193668*x^3-155456858376*x^2-422808875280*x
 +320938557273

Q3 = polredbest(Q3)
 -13*x^18+126*x^17-228*x^16+220*x^15+24*x^14
 -396*x^13+521*x^12-216*x^11-48*x^10-32*x^9-66*x^8
 +666*x^7-1013*x^6+348*x^5+510*x^4-654*x^3+234*x^2
 +36*x+9

gal = galoisinit(Q3);

gal.gen
 4,23,22,3,21,1,24,18,16,15,20]),Vecsmall([14,10,5,
 19,3,24,11,16,22,2,7,20,17,1,21,8,13,23,4,12,15,9,
 18,6]),Vecsmall([5,15,6,13,20,19,23,7,11,18,21,4,
 12,17,16,2,24,22,3,1,9,10,8,14]),Vecsmall([2,1,9,
 10,16,21,14,17,3,4,19,18,22,7,20,5,8,12,11,15,6,
 13,24,23])]

ord = gal.orders
prod(i=1,#ord,ord[i])

galoisidentify(gal)

L = galoissubgroups(gal);
#L

R1 = galoisfixedfield(gal,L[25])[1];
polgalois(R1)
R2 = galoisfixedfield(gal,L[28])[1];
polgalois(R2)

nf = nfinit(Q3);
factor(nf.disc)
[ 3 28]
[11 16]

dec3 = idealprimedec(nf,3);
pr3 = dec3[1];
[#dec3, pr3.f, pr3.e]

ram3 = idealramgroups(nf,gal,pr3);
#ram3

galoisidentify(ram3[1])
galoisisabelian(ram3[1])

galoisidentify(ram3[2])

galoisidentify(ram3[3])

dec11 = idealprimedec(nf,11);
pr11 = dec11[1];
[#dec11, pr11.f, pr11.e]

ram11 = idealramgroups(nf,gal,pr11);
#ram11

galoisidentify(ram11[1])
galoisidentify(ram11[2])

dec2 = idealprimedec(nf,2);
pr2 = dec2[1];
[#dec2, pr2.f, pr2.e]
frob2 = idealfrobenius(nf,gal,pr2);
permorder(frob2)

N = 7*13*19;
L1 = polsubcyclo(N,3);

L2 = [P | P <- L1, #factor(nfinit(P).disc)[,1]==3]
  x^3+x^2-576*x-5251, x^3+x^2-576*x+1665]

G = znstar(N)
  Mod(248, 1729), Mod(407, 1729)]]

H = mathnfmodid([1,0;-1,1;0,-1],3);

pol = galoissubcyclo(G,H)
factor(nfinit(pol).disc)
[ 7 2]
[13 2]
[19 2]

bnf = bnfinit(a^2-a+50);
bnf.cyc

R = bnrclassfield(bnf)[1]
 + (-12*a + 6)*x^4 - 30*x^3 + (18*a - 9)*x^2
 + 18*x + (-2*a + 1)

[cond,bnr,subg] = rnfconductor(bnf,R);
cond
subg

R2 = bnrclassfield(bnf,,2)
  + 244113*x^10 - 19818*x^8 - 3170*x^6
  + 17427*x^4 - 3258*x^2 + 199

bnr = bnrinit(bnf,12);
bnr.cyc

[deg,r1,D] = bnrdisc(bnr);
[deg,r1]
D

bnrclassfield(bnr)
 + (-294*a-3273)*x^4 + (-3*a-3852)*x^2 - 3,
 x^9 - 24*x^7 + (2*a-1)*x^6 + 495*x^5
 + (-12*a+6)*x^4 - 30*x^3 + (18*a-9)*x^2
 + 18*x + (-2*a+1)]

bnrclassfield(bnr,,1)

bnr = bnrinit(bnf,7)
bnr.cyc
bnrclassfield(bnr,3) \\elementary 3-subextension
 x^3 + (-1008*a - 651)*x + (-1103067*a - 8072813)]

pr41 = idealprimedec(bnf,41)[1];
bnrisprincipal(bnr,pr41,0)

bnr = bnrinit(bnf,[102709,43512;0,1]);
bnr.cyc
bnrclassfield(bnr,[9,3;0,1]) \\subgroup of index 9

bnf=bnfinit(a^2-217);
bnf.cyc
bnrinit(bnf,1).cyc
bnrinit(bnf,[1,[1,1]]).cyc

quadhilbert(-31)
lift(quadray(13,7))
  + (-91*y - 118)

bnf = bnfinit(x^2+2*3*5*7*11);
bnf.cyc
bnr = bnrinit(bnf,1,1);
gal = galoisinit(bnf);
m = bnrgaloismatrix(bnr,gal)[1]
[3 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]