// Magma code for working with number field 44.0.259678733580921933356150608489499806500103851346446927006805499909939891947499235044364762061824369642675407609.1.
// 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 + 126*x^42 - 136*x^41 + 6870*x^40 - 8230*x^39 + 213665*x^38 - 295965*x^37 + 4258263*x^36 - 7217913*x^35 + 60026803*x^34 - 132205933*x^33 + 702851619*x^32 - 2024910949*x^31 + 8478238424*x^30 - 28727347914*x^29 + 113007322201*x^28 - 400280801341*x^27 + 1543961581316*x^26 - 5546769594726*x^25 + 20996733686796*x^24 - 76464429634056*x^23 + 286238528671271*x^22 + 5510450813491562*x^21 - 2647951180883304*x^20 - 48120843156065008*x^19 + 21642532250445568*x^18 + 239910860018829312*x^17 - 23487886634093632*x^16 - 547804497609728384*x^15 + 312925302716625664*x^14 + 1663850613764854272*x^13 + 1465407373648837632*x^12 + 2924193837791471616*x^11 + 11729874814901612544*x^10 + 30257880986074456064*x^9 + 87040868574618779648*x^8 + 204532647457239990272*x^7 + 665876038859072012288*x^6 + 1376478266723886563328*x^5 + 5282282121480903327744*x^4 + 8468742695712451461120*x^3 + 44354078519157598453760*x^2 + 40322155800703304466432*x + 403218629390870461284352);

// 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 + 126*x^42 - 136*x^41 + 6870*x^40 - 8230*x^39 + 213665*x^38 - 295965*x^37 + 4258263*x^36 - 7217913*x^35 + 60026803*x^34 - 132205933*x^33 + 702851619*x^32 - 2024910949*x^31 + 8478238424*x^30 - 28727347914*x^29 + 113007322201*x^28 - 400280801341*x^27 + 1543961581316*x^26 - 5546769594726*x^25 + 20996733686796*x^24 - 76464429634056*x^23 + 286238528671271*x^22 + 5510450813491562*x^21 - 2647951180883304*x^20 - 48120843156065008*x^19 + 21642532250445568*x^18 + 239910860018829312*x^17 - 23487886634093632*x^16 - 547804497609728384*x^15 + 312925302716625664*x^14 + 1663850613764854272*x^13 + 1465407373648837632*x^12 + 2924193837791471616*x^11 + 11729874814901612544*x^10 + 30257880986074456064*x^9 + 87040868574618779648*x^8 + 204532647457239990272*x^7 + 665876038859072012288*x^6 + 1376478266723886563328*x^5 + 5282282121480903327744*x^4 + 8468742695712451461120*x^3 + 44354078519157598453760*x^2 + 40322155800703304466432*x + 403218629390870461284352);
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))];