// Magma code for working with number field 39.39.1112962024555065990379787974028986797706599025588599389261176471461970163669515376929.1

// 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 - 55*x^37 + 838*x^36 + 433*x^35 - 29490*x^34 + 36492*x^33 + 574075*x^32 - 1308560*x^31 - 6805710*x^30 + 21722262*x^29 + 50319569*x^28 - 220059875*x^27 - 219014481*x^26 + 1482194078*x^25 + 368114118*x^24 - 6887826619*x^23 + 1579984745*x^22 + 22412813436*x^21 - 12656284391*x^20 - 51075711901*x^19 + 41988060380*x^18 + 80500790810*x^17 - 83924117638*x^16 - 85491405725*x^15 + 107935481554*x^14 + 58560262932*x^13 - 89832821948*x^12 - 24031853409*x^11 + 47511398377*x^10 + 5015492400*x^9 - 15459668136*x^8 - 163687701*x^7 + 2936606920*x^6 - 133518320*x^5 - 294156298*x^4 + 23663051*x^3 + 11997840*x^2 - 1233420*x - 17513);

// 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 - 55*x^37 + 838*x^36 + 433*x^35 - 29490*x^34 + 36492*x^33 + 574075*x^32 - 1308560*x^31 - 6805710*x^30 + 21722262*x^29 + 50319569*x^28 - 220059875*x^27 - 219014481*x^26 + 1482194078*x^25 + 368114118*x^24 - 6887826619*x^23 + 1579984745*x^22 + 22412813436*x^21 - 12656284391*x^20 - 51075711901*x^19 + 41988060380*x^18 + 80500790810*x^17 - 83924117638*x^16 - 85491405725*x^15 + 107935481554*x^14 + 58560262932*x^13 - 89832821948*x^12 - 24031853409*x^11 + 47511398377*x^10 + 5015492400*x^9 - 15459668136*x^8 - 163687701*x^7 + 2936606920*x^6 - 133518320*x^5 - 294156298*x^4 + 23663051*x^3 + 11997840*x^2 - 1233420*x - 17513);
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))];