// Magma code for working with number field 44.44.181375764426442332776050749828434842919201892356144879261148162186145782470703125.1

// 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 - 44*x^42 + 43*x^41 + 902*x^40 - 859*x^39 - 11441*x^38 + 10582*x^37 + 100567*x^36 - 89985*x^35 - 650236*x^34 + 560251*x^33 + 3203650*x^32 - 2643399*x^31 - 12294569*x^30 + 9651170*x^29 + 37253225*x^28 - 27602055*x^27 - 89808680*x^26 + 62206625*x^25 + 172779011*x^24 - 110572386*x^23 - 265002643*x^22 + 154430258*x^21 + 322457859*x^20 - 168027624*x^19 - 308473797*x^18 + 140446403*x^17 + 228768475*x^16 - 88323383*x^15 - 128871012*x^14 + 40552321*x^13 + 53558102*x^12 - 13016729*x^11 - 15742859*x^10 + 2742874*x^9 + 3073662*x^8 - 347233*x^7 - 361455*x^6 + 24089*x^5 + 21786*x^4 - 986*x^3 - 504*x^2 + 24*x + 1);

// 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 - 44*x^42 + 43*x^41 + 902*x^40 - 859*x^39 - 11441*x^38 + 10582*x^37 + 100567*x^36 - 89985*x^35 - 650236*x^34 + 560251*x^33 + 3203650*x^32 - 2643399*x^31 - 12294569*x^30 + 9651170*x^29 + 37253225*x^28 - 27602055*x^27 - 89808680*x^26 + 62206625*x^25 + 172779011*x^24 - 110572386*x^23 - 265002643*x^22 + 154430258*x^21 + 322457859*x^20 - 168027624*x^19 - 308473797*x^18 + 140446403*x^17 + 228768475*x^16 - 88323383*x^15 - 128871012*x^14 + 40552321*x^13 + 53558102*x^12 - 13016729*x^11 - 15742859*x^10 + 2742874*x^9 + 3073662*x^8 - 347233*x^7 - 361455*x^6 + 24089*x^5 + 21786*x^4 - 986*x^3 - 504*x^2 + 24*x + 1);
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))];