// Magma code for working with number field 39.39.156428429087384921728870222525148611738414909735182186340401656558510812088760819892280320561915969.1

// Some of these functions may take a long time to execute (this depends on the field).

// Define the number field: 
R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^39 - 10*x^38 - 163*x^37 + 1930*x^36 + 10036*x^35 - 159252*x^34 - 244026*x^33 + 7427095*x^32 - 2389017*x^31 - 217511108*x^30 + 326735317*x^29 + 4189088926*x^28 - 10153118781*x^27 - 53678089242*x^26 + 179337849315*x^25 + 446153115950*x^24 - 2052054489049*x^23 - 2151039397974*x^22 + 15813321949956*x^21 + 2821647348255*x^20 - 82489984035674*x^19 + 33701919033939*x^18 + 286106777644803*x^17 - 243663734729882*x^16 - 630941672277082*x^15 + 788875500248585*x^14 + 805762979991898*x^13 - 1416423805872644*x^12 - 462194605752742*x^11 + 1418910665071105*x^10 - 50123800376408*x^9 - 755859394498746*x^8 + 182503268720113*x^7 + 191800233365760*x^6 - 72017807491341*x^5 - 18496135085754*x^4 + 9851455470329*x^3 + 124863566729*x^2 - 388373575025*x + 27293535527);

// Defining polynomial: 
DefiningPolynomial(K);

// Degree over Q: 
Degree(K);

// Signature: 
Signature(K);

// Discriminant: 
OK := Integers(K); Discriminant(OK);

// Ramified primes: 
PrimeDivisors(Discriminant(OK));

// Autmorphisms: 
Automorphisms(K);

// Integral basis: 
IntegralBasis(K);

// Class group: 
ClassGroup(K);

// Unit group: 
UK, fUK := UnitGroup(K);

// Unit rank: 
UnitRank(K);

// Generator for roots of unity: 
K!f(TU.1) where TU,f is TorsionUnitGroup(K);

// Fundamental units: 
[K|fUK(g): g in Generators(UK)];

// Regulator: 
Regulator(K);

// Analytic class number formula: 
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^39 - 10*x^38 - 163*x^37 + 1930*x^36 + 10036*x^35 - 159252*x^34 - 244026*x^33 + 7427095*x^32 - 2389017*x^31 - 217511108*x^30 + 326735317*x^29 + 4189088926*x^28 - 10153118781*x^27 - 53678089242*x^26 + 179337849315*x^25 + 446153115950*x^24 - 2052054489049*x^23 - 2151039397974*x^22 + 15813321949956*x^21 + 2821647348255*x^20 - 82489984035674*x^19 + 33701919033939*x^18 + 286106777644803*x^17 - 243663734729882*x^16 - 630941672277082*x^15 + 788875500248585*x^14 + 805762979991898*x^13 - 1416423805872644*x^12 - 462194605752742*x^11 + 1418910665071105*x^10 - 50123800376408*x^9 - 755859394498746*x^8 + 182503268720113*x^7 + 191800233365760*x^6 - 72017807491341*x^5 - 18496135085754*x^4 + 9851455470329*x^3 + 124863566729*x^2 - 388373575025*x + 27293535527);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

// Intermediate fields: 
L := Subfields(K); L[2..#L];

// Galois group: 
G = GaloisGroup(K);

// Frobenius cycle types: 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];