// Magma code for working with number field 32.32.71166659227026865952165739269552697151542080277732014965436610606901057.1

// 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 - 93*x^30 + 210*x^29 + 3538*x^28 - 12479*x^27 - 65768*x^26 + 350581*x^25 + 485064*x^24 - 5339623*x^23 + 2808135*x^22 + 44434017*x^21 - 84476060*x^20 - 170361963*x^19 + 681732194*x^18 - 77972047*x^17 - 2529499095*x^16 + 3133974961*x^15 + 3455889877*x^14 - 10578598738*x^13 + 3732760107*x^12 + 12702259726*x^11 - 15151938680*x^10 - 638400055*x^9 + 11610249674*x^8 - 6566502528*x^7 - 1290174274*x^6 + 2262714296*x^5 - 382096988*x^4 - 214409519*x^3 + 59339724*x^2 + 2830728*x + 14969);

// 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 - 93*x^30 + 210*x^29 + 3538*x^28 - 12479*x^27 - 65768*x^26 + 350581*x^25 + 485064*x^24 - 5339623*x^23 + 2808135*x^22 + 44434017*x^21 - 84476060*x^20 - 170361963*x^19 + 681732194*x^18 - 77972047*x^17 - 2529499095*x^16 + 3133974961*x^15 + 3455889877*x^14 - 10578598738*x^13 + 3732760107*x^12 + 12702259726*x^11 - 15151938680*x^10 - 638400055*x^9 + 11610249674*x^8 - 6566502528*x^7 - 1290174274*x^6 + 2262714296*x^5 - 382096988*x^4 - 214409519*x^3 + 59339724*x^2 + 2830728*x + 14969);
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))];