// Magma code for working with number field 44.0.3607583819545152459027384276140645884702253651640471494457079112798455926939134201954304.4.
// 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 - 2*x^43 + x^42 + 4*x^41 + 150*x^40 - 312*x^39 + 542*x^38 - 148*x^37 + 10492*x^36 - 20540*x^35 + 59284*x^34 - 56948*x^33 + 509181*x^32 - 847518*x^31 + 3165221*x^30 - 3787888*x^29 + 19028237*x^28 - 26692810*x^27 + 105670141*x^26 - 131261852*x^25 + 525362877*x^24 - 656940198*x^23 + 2435275925*x^22 - 2899731254*x^21 + 10236234869*x^20 - 11773277034*x^19 + 39267998773*x^18 - 43216075824*x^17 + 136381343684*x^16 - 143117336230*x^15 + 425588997760*x^14 - 421782473222*x^13 + 1177823357679*x^12 - 1090649084292*x^11 + 2847578340843*x^10 - 2421646596474*x^9 + 5860415317602*x^8 - 4459787108112*x^7 + 9896004307466*x^6 - 6407408568958*x^5 + 12768663644816*x^4 - 6318592552612*x^3 + 11065288394632*x^2 - 3173834791936*x + 4760716702561);

// 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 - 2*x^43 + x^42 + 4*x^41 + 150*x^40 - 312*x^39 + 542*x^38 - 148*x^37 + 10492*x^36 - 20540*x^35 + 59284*x^34 - 56948*x^33 + 509181*x^32 - 847518*x^31 + 3165221*x^30 - 3787888*x^29 + 19028237*x^28 - 26692810*x^27 + 105670141*x^26 - 131261852*x^25 + 525362877*x^24 - 656940198*x^23 + 2435275925*x^22 - 2899731254*x^21 + 10236234869*x^20 - 11773277034*x^19 + 39267998773*x^18 - 43216075824*x^17 + 136381343684*x^16 - 143117336230*x^15 + 425588997760*x^14 - 421782473222*x^13 + 1177823357679*x^12 - 1090649084292*x^11 + 2847578340843*x^10 - 2421646596474*x^9 + 5860415317602*x^8 - 4459787108112*x^7 + 9896004307466*x^6 - 6407408568958*x^5 + 12768663644816*x^4 - 6318592552612*x^3 + 11065288394632*x^2 - 3173834791936*x + 4760716702561);
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))];