// Magma code for working with number field 44.0.1829975953789394019358992112190249409734798500798529867331745089665450995586633384881.1

// 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 + 5*x^42 - 9*x^41 + 29*x^40 - 65*x^39 + 181*x^38 - 441*x^37 + 1165*x^36 - 2929*x^35 + 7589*x^34 - 19305*x^33 + 49661*x^32 - 126881*x^31 + 325525*x^30 - 833049*x^29 + 2135149*x^28 - 5467345*x^27 + 14007941*x^26 - 35877321*x^25 + 91909085*x^24 - 235418369*x^23 + 603054709*x^22 + 941673476*x^21 + 1470545360*x^20 + 2296148544*x^19 + 3586032896*x^18 + 5598561280*x^17 + 8745570304*x^16 + 13648674816*x^15 + 21333606400*x^14 + 33261092864*x^13 + 52073332736*x^12 + 80971038720*x^11 + 127322292224*x^10 + 196561862656*x^9 + 312727306240*x^8 + 473520144384*x^7 + 777389080576*x^6 + 1116691496960*x^5 + 1992864825344*x^4 + 2473901162496*x^3 + 5497558138880*x^2 + 4398046511104*x + 17592186044416);

// 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 + 5*x^42 - 9*x^41 + 29*x^40 - 65*x^39 + 181*x^38 - 441*x^37 + 1165*x^36 - 2929*x^35 + 7589*x^34 - 19305*x^33 + 49661*x^32 - 126881*x^31 + 325525*x^30 - 833049*x^29 + 2135149*x^28 - 5467345*x^27 + 14007941*x^26 - 35877321*x^25 + 91909085*x^24 - 235418369*x^23 + 603054709*x^22 + 941673476*x^21 + 1470545360*x^20 + 2296148544*x^19 + 3586032896*x^18 + 5598561280*x^17 + 8745570304*x^16 + 13648674816*x^15 + 21333606400*x^14 + 33261092864*x^13 + 52073332736*x^12 + 80971038720*x^11 + 127322292224*x^10 + 196561862656*x^9 + 312727306240*x^8 + 473520144384*x^7 + 777389080576*x^6 + 1116691496960*x^5 + 1992864825344*x^4 + 2473901162496*x^3 + 5497558138880*x^2 + 4398046511104*x + 17592186044416);
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))];