// Magma code for working with number field 32.0.2294081869728198830572049522437155246734619140625.1

// 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 - 11*x^31 + 68*x^30 - 336*x^29 + 1315*x^28 - 4335*x^27 + 12498*x^26 - 30713*x^25 + 67152*x^24 - 127199*x^23 + 211471*x^22 - 307078*x^21 + 387026*x^20 - 419617*x^19 + 359728*x^18 - 149755*x^17 - 229142*x^16 + 581095*x^15 - 441722*x^14 - 315448*x^13 + 1021466*x^12 - 946897*x^11 + 366311*x^10 - 50246*x^9 + 132237*x^8 - 215882*x^7 + 157773*x^6 - 97500*x^5 + 79930*x^4 - 54894*x^3 + 25063*x^2 - 7709*x + 3131);

// 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 - 11*x^31 + 68*x^30 - 336*x^29 + 1315*x^28 - 4335*x^27 + 12498*x^26 - 30713*x^25 + 67152*x^24 - 127199*x^23 + 211471*x^22 - 307078*x^21 + 387026*x^20 - 419617*x^19 + 359728*x^18 - 149755*x^17 - 229142*x^16 + 581095*x^15 - 441722*x^14 - 315448*x^13 + 1021466*x^12 - 946897*x^11 + 366311*x^10 - 50246*x^9 + 132237*x^8 - 215882*x^7 + 157773*x^6 - 97500*x^5 + 79930*x^4 - 54894*x^3 + 25063*x^2 - 7709*x + 3131);
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))];