// Magma code for working with number field 38.38.249822358047761737585176935673663749247824265938007663448506628820714951323582836933135986328125.1

// 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 - 17*x^37 - 64*x^36 + 2589*x^35 - 5411*x^34 - 156768*x^33 + 733103*x^32 + 4788192*x^31 - 35074270*x^30 - 69778288*x^29 + 937716801*x^28 - 3988006*x^27 - 15662818901*x^26 + 18940301798*x^25 + 169186329709*x^24 - 366424944806*x^23 - 1167394423359*x^22 + 3780063487716*x^21 + 4693995261962*x^20 - 24515638087910*x^19 - 6315614627194*x^18 + 103828189076722*x^17 - 36155927083660*x^16 - 285274970106320*x^15 + 223944436674986*x^14 + 486329219734937*x^13 - 571966582999537*x^12 - 458764738319995*x^11 + 781485719820911*x^10 + 160103399248715*x^9 - 557488639455834*x^8 + 53135732479684*x^7 + 180915511892981*x^6 - 38184876354047*x^5 - 25486623807243*x^4 + 6173666606950*x^3 + 1214949029058*x^2 - 273982926995*x + 184740541);

// 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 - 17*x^37 - 64*x^36 + 2589*x^35 - 5411*x^34 - 156768*x^33 + 733103*x^32 + 4788192*x^31 - 35074270*x^30 - 69778288*x^29 + 937716801*x^28 - 3988006*x^27 - 15662818901*x^26 + 18940301798*x^25 + 169186329709*x^24 - 366424944806*x^23 - 1167394423359*x^22 + 3780063487716*x^21 + 4693995261962*x^20 - 24515638087910*x^19 - 6315614627194*x^18 + 103828189076722*x^17 - 36155927083660*x^16 - 285274970106320*x^15 + 223944436674986*x^14 + 486329219734937*x^13 - 571966582999537*x^12 - 458764738319995*x^11 + 781485719820911*x^10 + 160103399248715*x^9 - 557488639455834*x^8 + 53135732479684*x^7 + 180915511892981*x^6 - 38184876354047*x^5 - 25486623807243*x^4 + 6173666606950*x^3 + 1214949029058*x^2 - 273982926995*x + 184740541);
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))];