// Magma code for working with number field 44.0.20914037845125665560658412713882054850915584464670209499044927044350221491820932802904570119521.1.
// 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 - 47*x^41 + 861*x^40 - 909*x^39 + 8208*x^38 - 9166*x^37 + 38346*x^36 - 48520*x^35 + 33092*x^34 - 92845*x^33 - 483814*x^32 + 301250*x^31 - 1903484*x^30 + 1624264*x^29 - 107439*x^28 - 19350*x^27 + 14282253*x^26 - 7869206*x^25 + 22479699*x^24 + 27193800*x^23 - 87279976*x^22 + 193887057*x^21 - 236841442*x^20 + 219628871*x^19 + 918204431*x^18 - 1297421237*x^17 + 3629171747*x^16 - 3180664266*x^15 - 485469968*x^14 - 3944578040*x^13 - 6043965750*x^12 - 31481752413*x^11 + 24001116410*x^10 - 15672532225*x^9 + 48314295842*x^8 + 34823525541*x^7 + 106431095722*x^6 - 4322349741*x^5 + 25050663427*x^4 - 162511397008*x^3 + 41768164772*x^2 - 9684104841*x + 24584641771);

// 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 + 46*x^42 - 47*x^41 + 861*x^40 - 909*x^39 + 8208*x^38 - 9166*x^37 + 38346*x^36 - 48520*x^35 + 33092*x^34 - 92845*x^33 - 483814*x^32 + 301250*x^31 - 1903484*x^30 + 1624264*x^29 - 107439*x^28 - 19350*x^27 + 14282253*x^26 - 7869206*x^25 + 22479699*x^24 + 27193800*x^23 - 87279976*x^22 + 193887057*x^21 - 236841442*x^20 + 219628871*x^19 + 918204431*x^18 - 1297421237*x^17 + 3629171747*x^16 - 3180664266*x^15 - 485469968*x^14 - 3944578040*x^13 - 6043965750*x^12 - 31481752413*x^11 + 24001116410*x^10 - 15672532225*x^9 + 48314295842*x^8 + 34823525541*x^7 + 106431095722*x^6 - 4322349741*x^5 + 25050663427*x^4 - 162511397008*x^3 + 41768164772*x^2 - 9684104841*x + 24584641771);
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))];