// Magma code for working with number field 32.0.3063491324247901158147277924095178749079926649475132560984974419980910475284097.1

// 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 + 100*x^30 - 369*x^29 + 4310*x^28 - 25796*x^27 + 146725*x^26 - 871109*x^25 + 4491744*x^24 - 22080636*x^23 + 104962263*x^22 - 459438417*x^21 + 1855235314*x^20 - 6919013614*x^19 + 23969357883*x^18 - 77412047603*x^17 + 232272715410*x^16 - 647554500300*x^15 + 1688255459920*x^14 - 4173402576653*x^13 + 9985317638552*x^12 - 23194926103486*x^11 + 50683905711002*x^10 - 99818909359402*x^9 + 174984915122456*x^8 - 284753440308162*x^7 + 455392827342446*x^6 - 694285600401985*x^5 + 894386134923647*x^4 - 862139564170835*x^3 + 562130389554400*x^2 - 218583702854457*x + 38241539870119);

// 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 + 100*x^30 - 369*x^29 + 4310*x^28 - 25796*x^27 + 146725*x^26 - 871109*x^25 + 4491744*x^24 - 22080636*x^23 + 104962263*x^22 - 459438417*x^21 + 1855235314*x^20 - 6919013614*x^19 + 23969357883*x^18 - 77412047603*x^17 + 232272715410*x^16 - 647554500300*x^15 + 1688255459920*x^14 - 4173402576653*x^13 + 9985317638552*x^12 - 23194926103486*x^11 + 50683905711002*x^10 - 99818909359402*x^9 + 174984915122456*x^8 - 284753440308162*x^7 + 455392827342446*x^6 - 694285600401985*x^5 + 894386134923647*x^4 - 862139564170835*x^3 + 562130389554400*x^2 - 218583702854457*x + 38241539870119);
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))];