// Magma code for working with number field 44.0.116567320065927752512435466812933331534234135648947894549180382317450046539306640625.3

// 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 - 3*x^42 + 7*x^41 + 5*x^40 - 33*x^39 + 13*x^38 + 119*x^37 - 171*x^36 - 305*x^35 + 989*x^34 + 231*x^33 - 4187*x^32 + 3263*x^31 + 13485*x^30 - 26537*x^29 - 27403*x^28 + 133551*x^27 - 23939*x^26 - 510265*x^25 + 606021*x^24 + 1435039*x^23 - 3859123*x^22 + 5740156*x^21 + 9696336*x^20 - 32656960*x^19 - 6128384*x^18 + 136756224*x^17 - 112242688*x^16 - 434782208*x^15 + 883752960*x^14 + 855375872*x^13 - 4390387712*x^12 + 968884224*x^11 + 16592666624*x^10 - 20468203520*x^9 - 45902462976*x^8 + 127775277056*x^7 + 55834574848*x^6 - 566935683072*x^5 + 343597383680*x^4 + 1924145348608*x^3 - 3298534883328*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 - 3*x^42 + 7*x^41 + 5*x^40 - 33*x^39 + 13*x^38 + 119*x^37 - 171*x^36 - 305*x^35 + 989*x^34 + 231*x^33 - 4187*x^32 + 3263*x^31 + 13485*x^30 - 26537*x^29 - 27403*x^28 + 133551*x^27 - 23939*x^26 - 510265*x^25 + 606021*x^24 + 1435039*x^23 - 3859123*x^22 + 5740156*x^21 + 9696336*x^20 - 32656960*x^19 - 6128384*x^18 + 136756224*x^17 - 112242688*x^16 - 434782208*x^15 + 883752960*x^14 + 855375872*x^13 - 4390387712*x^12 + 968884224*x^11 + 16592666624*x^10 - 20468203520*x^9 - 45902462976*x^8 + 127775277056*x^7 + 55834574848*x^6 - 566935683072*x^5 + 343597383680*x^4 + 1924145348608*x^3 - 3298534883328*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))];