// Magma code for working with number field 44.0.281428024862179713811519790826061407189965383375188019446717490122860358656599250463494607518141376885333.1.
// 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 + 188*x^42 - 191*x^41 + 15682*x^40 - 16255*x^39 + 767101*x^38 - 815866*x^37 + 24539323*x^36 - 26986921*x^35 + 543368874*x^34 - 624329637*x^33 + 8632506022*x^32 - 10505494933*x^31 + 101311538399*x^30 - 132828023198*x^29 + 907079143955*x^28 - 1305563213549*x^27 + 6469389498694*x^26 - 10386079139341*x^25 + 38961713404851*x^24 - 70119950822874*x^23 + 212107487819539*x^22 - 267306271081404*x^21 + 954204965731533*x^20 - 348452808851934*x^19 + 3286452427237125*x^18 + 1220729003449623*x^17 + 8721817319211705*x^16 + 7424451290225763*x^15 + 18808628124207888*x^14 + 21088254372919965*x^13 + 35377638646744746*x^12 + 44100561928942881*x^11 + 62049209117584005*x^10 + 79953228494360262*x^9 + 106199279503507242*x^8 + 136817944804830789*x^7 + 181780810904989953*x^6 + 226679666465622069*x^5 + 318662867727691812*x^4 + 354165989590339128*x^3 + 601822619090434404*x^2 + 452107998974921166*x + 1353359858390092809);

// Defining polynomial: 
DefiningPolynomial(K);

// Degree over Q: 
Degree(K);

// Signature: 
Signature(K);

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

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

// Automorphisms: 
Automorphisms(K);

// Integral basis: 
IntegralBasis(K);

// Class group: 
ClassGroup(K);

// Narrow class group: 
NarrowClassGroup(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(Rationals()); K<a> := NumberField(x^44 - x^43 + 188*x^42 - 191*x^41 + 15682*x^40 - 16255*x^39 + 767101*x^38 - 815866*x^37 + 24539323*x^36 - 26986921*x^35 + 543368874*x^34 - 624329637*x^33 + 8632506022*x^32 - 10505494933*x^31 + 101311538399*x^30 - 132828023198*x^29 + 907079143955*x^28 - 1305563213549*x^27 + 6469389498694*x^26 - 10386079139341*x^25 + 38961713404851*x^24 - 70119950822874*x^23 + 212107487819539*x^22 - 267306271081404*x^21 + 954204965731533*x^20 - 348452808851934*x^19 + 3286452427237125*x^18 + 1220729003449623*x^17 + 8721817319211705*x^16 + 7424451290225763*x^15 + 18808628124207888*x^14 + 21088254372919965*x^13 + 35377638646744746*x^12 + 44100561928942881*x^11 + 62049209117584005*x^10 + 79953228494360262*x^9 + 106199279503507242*x^8 + 136817944804830789*x^7 + 181780810904989953*x^6 + 226679666465622069*x^5 + 318662867727691812*x^4 + 354165989590339128*x^3 + 601822619090434404*x^2 + 452107998974921166*x + 1353359858390092809);
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 pO_K for p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];