// Magma code for working with number field 28.0.3654513548980364725264702136425345220781793212890625.1.
// 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^28 - 5*x^27 + 47*x^26 - 152*x^25 + 1049*x^24 - 3001*x^23 + 14492*x^22 - 30908*x^21 + 115818*x^20 - 204097*x^19 + 649914*x^18 - 933216*x^17 + 2508376*x^16 - 2978786*x^15 + 7071411*x^14 - 6804920*x^13 + 13577493*x^12 - 10099944*x^11 + 17701828*x^10 - 9450995*x^9 + 11808392*x^8 - 1974799*x^7 + 3396106*x^6 - 287718*x^5 + 726277*x^4 + 16080*x^3 + 60245*x^2 - 6888*x + 1681);

// 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(Rationals()); K<a> := NumberField(x^28 - 5*x^27 + 47*x^26 - 152*x^25 + 1049*x^24 - 3001*x^23 + 14492*x^22 - 30908*x^21 + 115818*x^20 - 204097*x^19 + 649914*x^18 - 933216*x^17 + 2508376*x^16 - 2978786*x^15 + 7071411*x^14 - 6804920*x^13 + 13577493*x^12 - 10099944*x^11 + 17701828*x^10 - 9450995*x^9 + 11808392*x^8 - 1974799*x^7 + 3396106*x^6 - 287718*x^5 + 726277*x^4 + 16080*x^3 + 60245*x^2 - 6888*x + 1681);
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 pO_K for p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];