// Magma code for working with number field 44.0.191158492466532603992882058813386761401107535956984758058028173625115327713733756218489.1

// 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^44 - x^43 + 46*x^42 - 45*x^41 + 1195*x^40 - 1150*x^39 + 21299*x^38 - 20149*x^37 + 287730*x^36 - 267581*x^35 + 3072523*x^34 - 2804942*x^33 + 26692099*x^32 - 23887157*x^31 + 191331250*x^30 - 167444093*x^29 + 1142837707*x^28 - 975393614*x^27 + 5704151939*x^26 - 4728758325*x^25 + 23804513458*x^24 - 19075755133*x^23 + 82643603403*x^22 - 63567853467*x^21 + 237057619920*x^20 - 173490244577*x^19 + 554044068881*x^18 - 380558605544*x^17 + 1037868132567*x^16 - 657255020887*x^15 + 1512341393536*x^14 - 854306074281*x^13 + 1655465958953*x^12 - 799339188480*x^11 + 1266628178647*x^10 - 472858178519*x^9 + 634019315968*x^8 - 183040091689*x^7 + 183593805097*x^6 - 13681085952*x^5 + 19238856919*x^4 + 3193810729*x^3 + 1973987328*x^2 + 217790679*x + 27008809);

// 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^44 - x^43 + 46*x^42 - 45*x^41 + 1195*x^40 - 1150*x^39 + 21299*x^38 - 20149*x^37 + 287730*x^36 - 267581*x^35 + 3072523*x^34 - 2804942*x^33 + 26692099*x^32 - 23887157*x^31 + 191331250*x^30 - 167444093*x^29 + 1142837707*x^28 - 975393614*x^27 + 5704151939*x^26 - 4728758325*x^25 + 23804513458*x^24 - 19075755133*x^23 + 82643603403*x^22 - 63567853467*x^21 + 237057619920*x^20 - 173490244577*x^19 + 554044068881*x^18 - 380558605544*x^17 + 1037868132567*x^16 - 657255020887*x^15 + 1512341393536*x^14 - 854306074281*x^13 + 1655465958953*x^12 - 799339188480*x^11 + 1266628178647*x^10 - 472858178519*x^9 + 634019315968*x^8 - 183040091689*x^7 + 183593805097*x^6 - 13681085952*x^5 + 19238856919*x^4 + 3193810729*x^3 + 1973987328*x^2 + 217790679*x + 27008809);
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))];