// Magma code for working with number field 44.0.6091524468845815638846997953624350524431889522210325630882624592675376276321.1

// 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 - x^42 + 3*x^41 - x^40 - 5*x^39 + 7*x^38 + 3*x^37 - 17*x^36 + 11*x^35 + 23*x^34 - 45*x^33 - x^32 + 91*x^31 - 89*x^30 - 93*x^29 + 271*x^28 - 85*x^27 - 457*x^26 + 627*x^25 + 287*x^24 - 1541*x^23 + 967*x^22 - 3082*x^21 + 1148*x^20 + 5016*x^19 - 7312*x^18 - 2720*x^17 + 17344*x^16 - 11904*x^15 - 22784*x^14 + 46592*x^13 - 1024*x^12 - 92160*x^11 + 94208*x^10 + 90112*x^9 - 278528*x^8 + 98304*x^7 + 458752*x^6 - 655360*x^5 - 262144*x^4 + 1572864*x^3 - 1048576*x^2 - 2097152*x + 4194304);

// 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 - x^42 + 3*x^41 - x^40 - 5*x^39 + 7*x^38 + 3*x^37 - 17*x^36 + 11*x^35 + 23*x^34 - 45*x^33 - x^32 + 91*x^31 - 89*x^30 - 93*x^29 + 271*x^28 - 85*x^27 - 457*x^26 + 627*x^25 + 287*x^24 - 1541*x^23 + 967*x^22 - 3082*x^21 + 1148*x^20 + 5016*x^19 - 7312*x^18 - 2720*x^17 + 17344*x^16 - 11904*x^15 - 22784*x^14 + 46592*x^13 - 1024*x^12 - 92160*x^11 + 94208*x^10 + 90112*x^9 - 278528*x^8 + 98304*x^7 + 458752*x^6 - 655360*x^5 - 262144*x^4 + 1572864*x^3 - 1048576*x^2 - 2097152*x + 4194304);
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))];