// Magma code for working with number field 38.38.2907625224541948887418190194159474648332884150922805055774449275194637293139188364462591928677.1

// 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^38 - x^37 - 188*x^36 + 180*x^35 + 15133*x^34 - 12829*x^33 - 695929*x^32 + 470210*x^31 + 20560148*x^30 - 9452291*x^29 - 414286793*x^28 + 90520176*x^27 + 5875939616*x^26 + 191303612*x^25 - 59496127537*x^24 - 16762796308*x^23 + 430644226744*x^22 + 224673611403*x^21 - 2206305371572*x^20 - 1655031822292*x^19 + 7828656065108*x^18 + 7607383848877*x^17 - 18551523651529*x^16 - 22355418258850*x^15 + 27686404325779*x^14 + 41724858463028*x^13 - 23225313872578*x^12 - 48637857116067*x^11 + 6837701957301*x^10 + 34223650023543*x^9 + 4693738691921*x^8 - 13444562674843*x^7 - 4790142874347*x^6 + 2352031466145*x^5 + 1507820068977*x^4 + 14043380009*x^3 - 154333274880*x^2 - 38062864399*x - 2836879451);

// 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^38 - x^37 - 188*x^36 + 180*x^35 + 15133*x^34 - 12829*x^33 - 695929*x^32 + 470210*x^31 + 20560148*x^30 - 9452291*x^29 - 414286793*x^28 + 90520176*x^27 + 5875939616*x^26 + 191303612*x^25 - 59496127537*x^24 - 16762796308*x^23 + 430644226744*x^22 + 224673611403*x^21 - 2206305371572*x^20 - 1655031822292*x^19 + 7828656065108*x^18 + 7607383848877*x^17 - 18551523651529*x^16 - 22355418258850*x^15 + 27686404325779*x^14 + 41724858463028*x^13 - 23225313872578*x^12 - 48637857116067*x^11 + 6837701957301*x^10 + 34223650023543*x^9 + 4693738691921*x^8 - 13444562674843*x^7 - 4790142874347*x^6 + 2352031466145*x^5 + 1507820068977*x^4 + 14043380009*x^3 - 154333274880*x^2 - 38062864399*x - 2836879451);
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))];