// Magma code for working with number field 44.0.361358208821422691858000111178424879775250540561407860223115640123091356736736779241.1

// 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 - 31*x^42 + 18*x^41 + 509*x^40 - 128*x^39 - 6323*x^38 + 711*x^37 + 64913*x^36 - 2855*x^35 - 548822*x^34 - 30450*x^33 + 3909507*x^32 + 497028*x^31 - 24053724*x^30 - 3501489*x^29 + 127000724*x^28 + 22049413*x^27 - 570173510*x^26 - 120800376*x^25 + 2191193723*x^24 + 441547756*x^23 - 7154579015*x^22 - 1235011630*x^21 + 19357719068*x^20 + 3692553824*x^19 - 42800540592*x^18 - 8560769600*x^17 + 77181753152*x^16 + 8780862080*x^15 - 109191557376*x^14 - 4336701952*x^13 + 115169783808*x^12 + 13857435648*x^11 - 86371995648*x^10 - 18961448960*x^9 + 47011856384*x^8 + 1711374336*x^7 - 14905901056*x^6 + 3448373248*x^5 + 3020947456*x^4 - 373293056*x^3 - 84934656*x^2 - 12582912*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 - 31*x^42 + 18*x^41 + 509*x^40 - 128*x^39 - 6323*x^38 + 711*x^37 + 64913*x^36 - 2855*x^35 - 548822*x^34 - 30450*x^33 + 3909507*x^32 + 497028*x^31 - 24053724*x^30 - 3501489*x^29 + 127000724*x^28 + 22049413*x^27 - 570173510*x^26 - 120800376*x^25 + 2191193723*x^24 + 441547756*x^23 - 7154579015*x^22 - 1235011630*x^21 + 19357719068*x^20 + 3692553824*x^19 - 42800540592*x^18 - 8560769600*x^17 + 77181753152*x^16 + 8780862080*x^15 - 109191557376*x^14 - 4336701952*x^13 + 115169783808*x^12 + 13857435648*x^11 - 86371995648*x^10 - 18961448960*x^9 + 47011856384*x^8 + 1711374336*x^7 - 14905901056*x^6 + 3448373248*x^5 + 3020947456*x^4 - 373293056*x^3 - 84934656*x^2 - 12582912*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))];