// Magma code for working with number field 32.32.2318482735674757180308401186646582923236113207344277714859125196933746337890625.2

// 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^32 - x^31 - 192*x^30 + 254*x^29 + 15176*x^28 - 22502*x^27 - 659185*x^26 + 1014228*x^25 + 17695139*x^24 - 27276312*x^23 - 311391664*x^22 + 475350824*x^21 + 3700993657*x^20 - 5615084090*x^19 - 30036394961*x^18 + 45970229245*x^17 + 165619980768*x^16 - 262233944896*x^15 - 606028450275*x^14 + 1031625598042*x^13 + 1394173531832*x^12 - 2722189078308*x^11 - 1767203506112*x^10 + 4578752971493*x^9 + 675907939625*x^8 - 4492953890543*x^7 + 922086899405*x^6 + 2162678325954*x^5 - 1005718811728*x^4 - 309785246453*x^3 + 244081450784*x^2 - 15465730875*x - 7522269649);

// 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^32 - x^31 - 192*x^30 + 254*x^29 + 15176*x^28 - 22502*x^27 - 659185*x^26 + 1014228*x^25 + 17695139*x^24 - 27276312*x^23 - 311391664*x^22 + 475350824*x^21 + 3700993657*x^20 - 5615084090*x^19 - 30036394961*x^18 + 45970229245*x^17 + 165619980768*x^16 - 262233944896*x^15 - 606028450275*x^14 + 1031625598042*x^13 + 1394173531832*x^12 - 2722189078308*x^11 - 1767203506112*x^10 + 4578752971493*x^9 + 675907939625*x^8 - 4492953890543*x^7 + 922086899405*x^6 + 2162678325954*x^5 - 1005718811728*x^4 - 309785246453*x^3 + 244081450784*x^2 - 15465730875*x - 7522269649);
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))];