// Magma code for working with number field 33.33.45897850273808078905473711375352471596627685094598222212495078792349891889.1

// 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 - 8*x^32 - 114*x^31 + 1061*x^30 + 5059*x^29 - 60653*x^28 - 96801*x^27 + 1959353*x^26 - 29225*x^25 - 39423204*x^24 + 40518048*x^23 + 513641811*x^22 - 918583926*x^21 - 4362858468*x^20 + 10893040871*x^19 + 23607826158*x^18 - 79952364670*x^17 - 75116228962*x^16 + 381639679930*x^15 + 98812353215*x^14 - 1205788951741*x^13 + 175885903114*x^12 + 2530475990857*x^11 - 987113141315*x^10 - 3498652499635*x^9 + 1841635007257*x^8 + 3110258033850*x^7 - 1787906886467*x^6 - 1685908305594*x^5 + 902138910730*x^4 + 495599994390*x^3 - 195854103442*x^2 - 57250484608*x + 8460460991);

// 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 - 8*x^32 - 114*x^31 + 1061*x^30 + 5059*x^29 - 60653*x^28 - 96801*x^27 + 1959353*x^26 - 29225*x^25 - 39423204*x^24 + 40518048*x^23 + 513641811*x^22 - 918583926*x^21 - 4362858468*x^20 + 10893040871*x^19 + 23607826158*x^18 - 79952364670*x^17 - 75116228962*x^16 + 381639679930*x^15 + 98812353215*x^14 - 1205788951741*x^13 + 175885903114*x^12 + 2530475990857*x^11 - 987113141315*x^10 - 3498652499635*x^9 + 1841635007257*x^8 + 3110258033850*x^7 - 1787906886467*x^6 - 1685908305594*x^5 + 902138910730*x^4 + 495599994390*x^3 - 195854103442*x^2 - 57250484608*x + 8460460991);
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))];