// Magma code for working with number field 44.44.2829456642779506738660199294300931896594438764890376623447387777021003184630861677846958804464951379972781.1.
// 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 - 245*x^42 + 238*x^41 + 27464*x^40 - 25798*x^39 - 1870216*x^38 + 1689630*x^37 + 86645247*x^36 - 74817837*x^35 - 2898974188*x^34 + 2375249329*x^33 + 72579947126*x^32 - 55953201823*x^31 - 1390382110630*x^30 + 998709697869*x^29 + 20665571127124*x^28 - 13674603242041*x^27 - 240238917307978*x^26 + 144516694613691*x^25 + 2191711606172907*x^24 - 1180094743877070*x^23 - 15680069546419541*x^22 + 7419430297867336*x^21 + 87555848961529704*x^20 - 35621383921138009*x^19 - 378178023714978041*x^18 + 128871770255036598*x^17 + 1246087932552175079*x^16 - 344680676241032240*x^15 - 3070152424219288766*x^14 + 664374849384426306*x^13 + 5499539488500745998*x^12 - 894737827553875328*x^11 - 6883781452983976207*x^10 + 818014176860906167*x^9 + 5684771260919185772*x^8 - 508187269302072418*x^7 - 2836917120327885752*x^6 + 230965518365555296*x^5 + 736177518063946879*x^4 - 79966127580583048*x^3 - 70130358520706481*x^2 + 13792867712635819*x - 669015421113803);

// 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 - 245*x^42 + 238*x^41 + 27464*x^40 - 25798*x^39 - 1870216*x^38 + 1689630*x^37 + 86645247*x^36 - 74817837*x^35 - 2898974188*x^34 + 2375249329*x^33 + 72579947126*x^32 - 55953201823*x^31 - 1390382110630*x^30 + 998709697869*x^29 + 20665571127124*x^28 - 13674603242041*x^27 - 240238917307978*x^26 + 144516694613691*x^25 + 2191711606172907*x^24 - 1180094743877070*x^23 - 15680069546419541*x^22 + 7419430297867336*x^21 + 87555848961529704*x^20 - 35621383921138009*x^19 - 378178023714978041*x^18 + 128871770255036598*x^17 + 1246087932552175079*x^16 - 344680676241032240*x^15 - 3070152424219288766*x^14 + 664374849384426306*x^13 + 5499539488500745998*x^12 - 894737827553875328*x^11 - 6883781452983976207*x^10 + 818014176860906167*x^9 + 5684771260919185772*x^8 - 508187269302072418*x^7 - 2836917120327885752*x^6 + 230965518365555296*x^5 + 736177518063946879*x^4 - 79966127580583048*x^3 - 70130358520706481*x^2 + 13792867712635819*x - 669015421113803);
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))];