// Magma code for working with number field 44.0.1555282614282872031686695202841759550456839851168205668482420621934906547732043786064001.1

// 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 - 82*x^41 + 77*x^40 + 73*x^39 + 2795*x^38 - 2452*x^37 - 2193*x^36 - 51847*x^35 + 42213*x^34 + 35713*x^33 + 576416*x^32 - 432884*x^31 - 345487*x^30 - 3995639*x^29 + 2790104*x^28 + 2010938*x^27 + 17436469*x^26 - 11948291*x^25 - 6590980*x^24 - 47156450*x^23 + 36319435*x^22 + 9637447*x^21 + 76421426*x^20 - 81395780*x^19 + 4175237*x^18 - 78460015*x^17 + 124870243*x^16 - 45435254*x^15 + 82436729*x^14 - 106904257*x^13 + 95684734*x^12 - 78566744*x^11 + 62868338*x^10 - 89632848*x^9 + 65796469*x^8 - 36457247*x^7 + 9020998*x^6 - 24902176*x^5 + 25739321*x^4 + 13163316*x^3 - 5336842*x^2 + 199578*x + 1151329);

// 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 - 82*x^41 + 77*x^40 + 73*x^39 + 2795*x^38 - 2452*x^37 - 2193*x^36 - 51847*x^35 + 42213*x^34 + 35713*x^33 + 576416*x^32 - 432884*x^31 - 345487*x^30 - 3995639*x^29 + 2790104*x^28 + 2010938*x^27 + 17436469*x^26 - 11948291*x^25 - 6590980*x^24 - 47156450*x^23 + 36319435*x^22 + 9637447*x^21 + 76421426*x^20 - 81395780*x^19 + 4175237*x^18 - 78460015*x^17 + 124870243*x^16 - 45435254*x^15 + 82436729*x^14 - 106904257*x^13 + 95684734*x^12 - 78566744*x^11 + 62868338*x^10 - 89632848*x^9 + 65796469*x^8 - 36457247*x^7 + 9020998*x^6 - 24902176*x^5 + 25739321*x^4 + 13163316*x^3 - 5336842*x^2 + 199578*x + 1151329);
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))];