// Magma code for working with number field 33.33.27189028279553414235049966267283185807800188603627566700161.1

// 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^33 - x^32 - 32*x^31 + 31*x^30 + 465*x^29 - 435*x^28 - 4060*x^27 + 3654*x^26 + 23751*x^25 - 20475*x^24 - 98280*x^23 + 80730*x^22 + 296010*x^21 - 230230*x^20 - 657800*x^19 + 480700*x^18 + 1081575*x^17 - 735471*x^16 - 1307504*x^15 + 817190*x^14 + 1144066*x^13 - 646646*x^12 - 705432*x^11 + 352716*x^10 + 293930*x^9 - 125970*x^8 - 77520*x^7 + 27132*x^6 + 11628*x^5 - 3060*x^4 - 816*x^3 + 136*x^2 + 17*x - 1);

// 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^33 - x^32 - 32*x^31 + 31*x^30 + 465*x^29 - 435*x^28 - 4060*x^27 + 3654*x^26 + 23751*x^25 - 20475*x^24 - 98280*x^23 + 80730*x^22 + 296010*x^21 - 230230*x^20 - 657800*x^19 + 480700*x^18 + 1081575*x^17 - 735471*x^16 - 1307504*x^15 + 817190*x^14 + 1144066*x^13 - 646646*x^12 - 705432*x^11 + 352716*x^10 + 293930*x^9 - 125970*x^8 - 77520*x^7 + 27132*x^6 + 11628*x^5 - 3060*x^4 - 816*x^3 + 136*x^2 + 17*x - 1);
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))];