// Magma code for working with number field 44.0.342865339180420288801608222738062084913425127327306009945459663867950439453125.1

// 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 + 11*x^42 - 12*x^41 + 77*x^40 - 74*x^39 + 424*x^38 - 368*x^37 + 2052*x^36 - 1675*x^35 + 8154*x^34 - 6259*x^33 + 27860*x^32 - 18614*x^31 + 82991*x^30 - 50150*x^29 + 218995*x^28 - 122605*x^27 + 487495*x^26 - 239465*x^25 + 942321*x^24 - 367121*x^23 + 1562162*x^22 - 556097*x^21 + 2230394*x^20 - 742219*x^19 + 2512113*x^18 - 607127*x^17 + 2369760*x^16 - 263798*x^15 + 1750478*x^14 - 273124*x^13 + 1076097*x^12 - 237409*x^11 + 402111*x^10 - 41911*x^9 + 123922*x^8 + 19642*x^7 + 17600*x^6 + 1439*x^5 + 2686*x^4 - 361*x^3 + 51*x^2 - 6*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 + 11*x^42 - 12*x^41 + 77*x^40 - 74*x^39 + 424*x^38 - 368*x^37 + 2052*x^36 - 1675*x^35 + 8154*x^34 - 6259*x^33 + 27860*x^32 - 18614*x^31 + 82991*x^30 - 50150*x^29 + 218995*x^28 - 122605*x^27 + 487495*x^26 - 239465*x^25 + 942321*x^24 - 367121*x^23 + 1562162*x^22 - 556097*x^21 + 2230394*x^20 - 742219*x^19 + 2512113*x^18 - 607127*x^17 + 2369760*x^16 - 263798*x^15 + 1750478*x^14 - 273124*x^13 + 1076097*x^12 - 237409*x^11 + 402111*x^10 - 41911*x^9 + 123922*x^8 + 19642*x^7 + 17600*x^6 + 1439*x^5 + 2686*x^4 - 361*x^3 + 51*x^2 - 6*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))];