// Magma code for working with number field 32.0.5935315803327378381589507037815252283484449810801350950039360504150390625.1

// 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^32 - x^31 + 99*x^30 + 157*x^29 + 3924*x^28 + 16492*x^27 + 103817*x^26 + 576079*x^25 + 2396881*x^24 + 11252767*x^23 + 43066027*x^22 + 154641664*x^21 + 531776968*x^20 + 1604710009*x^19 + 4592804568*x^18 + 12105474106*x^17 + 28991395516*x^16 + 65009104064*x^15 + 133385029876*x^14 + 251976154626*x^13 + 441515729309*x^12 + 705402557166*x^11 + 1044251596422*x^10 + 1417722437928*x^9 + 1747465221173*x^8 + 2010198542396*x^7 + 2043970195671*x^6 + 1742678971704*x^5 + 1178198932512*x^4 + 286794915794*x^3 - 161102822545*x^2 - 9076955370*x + 73651979561);

// 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 + 99*x^30 + 157*x^29 + 3924*x^28 + 16492*x^27 + 103817*x^26 + 576079*x^25 + 2396881*x^24 + 11252767*x^23 + 43066027*x^22 + 154641664*x^21 + 531776968*x^20 + 1604710009*x^19 + 4592804568*x^18 + 12105474106*x^17 + 28991395516*x^16 + 65009104064*x^15 + 133385029876*x^14 + 251976154626*x^13 + 441515729309*x^12 + 705402557166*x^11 + 1044251596422*x^10 + 1417722437928*x^9 + 1747465221173*x^8 + 2010198542396*x^7 + 2043970195671*x^6 + 1742678971704*x^5 + 1178198932512*x^4 + 286794915794*x^3 - 161102822545*x^2 - 9076955370*x + 73651979561);
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))];