// Magma code for working with number field 44.44.1588950548018177733817067321623614819860239312933539604872830957860279621432150312826633453369140625.1.
// 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 - 132*x^42 + 131*x^41 + 7981*x^40 - 7851*x^39 - 293502*x^38 + 285780*x^37 + 7352544*x^36 - 7074358*x^35 - 133171784*x^34 + 126368145*x^33 + 1806620424*x^32 - 1686827784*x^31 - 18764844858*x^30 + 17194056330*x^29 + 151268636295*x^28 - 135618356054*x^27 - 953714610433*x^26 + 833885620680*x^25 + 4717270351997*x^24 - 4008528403014*x^23 - 18288247142006*x^22 + 15048715494929*x^21 + 55312614944324*x^20 - 43911599382957*x^19 - 129393380339543*x^18 + 98733171381559*x^17 + 231114721012025*x^16 - 168838600437028*x^15 - 309586495847682*x^14 + 215575931040434*x^13 + 303617014545478*x^12 - 200381975387743*x^11 - 211144037309930*x^10 + 130979137701417*x^9 + 99728212374110*x^8 - 57361851429233*x^7 - 30095367088880*x^6 + 15665394240053*x^5 + 5275330691163*x^4 - 2357998728050*x^3 - 452559496938*x^2 + 146388138349*x + 12360161581);

// 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 - 132*x^42 + 131*x^41 + 7981*x^40 - 7851*x^39 - 293502*x^38 + 285780*x^37 + 7352544*x^36 - 7074358*x^35 - 133171784*x^34 + 126368145*x^33 + 1806620424*x^32 - 1686827784*x^31 - 18764844858*x^30 + 17194056330*x^29 + 151268636295*x^28 - 135618356054*x^27 - 953714610433*x^26 + 833885620680*x^25 + 4717270351997*x^24 - 4008528403014*x^23 - 18288247142006*x^22 + 15048715494929*x^21 + 55312614944324*x^20 - 43911599382957*x^19 - 129393380339543*x^18 + 98733171381559*x^17 + 231114721012025*x^16 - 168838600437028*x^15 - 309586495847682*x^14 + 215575931040434*x^13 + 303617014545478*x^12 - 200381975387743*x^11 - 211144037309930*x^10 + 130979137701417*x^9 + 99728212374110*x^8 - 57361851429233*x^7 - 30095367088880*x^6 + 15665394240053*x^5 + 5275330691163*x^4 - 2357998728050*x^3 - 452559496938*x^2 + 146388138349*x + 12360161581);
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))];