// Magma code for working with number field 24.4.6769025210045733840131040987091228089411742985248565673828125.6.
// 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^24 - 2*x^23 + 119*x^22 - 123*x^21 - 2284*x^20 + 35021*x^19 - 287932*x^18 - 3528451*x^17 + 430787*x^16 + 76267881*x^15 + 383091958*x^14 + 1547361734*x^13 + 1926262502*x^12 - 24636617534*x^11 - 181070920997*x^10 - 710772344063*x^9 - 1879981260494*x^8 - 2702189457572*x^7 + 3316968239109*x^6 + 29845357531527*x^5 + 75565793089194*x^4 + 102052357360777*x^3 + 77691878905336*x^2 + 30942834489368*x + 4794247932259);

// Defining polynomial: 
DefiningPolynomial(K);

// Degree over Q: 
Degree(K);

// Signature: 
Signature(K);

// Discriminant: 
OK := Integers(K); Discriminant(OK);

// Ramified primes: 
PrimeDivisors(Discriminant(OK));

// Automorphisms: 
Automorphisms(K);

// Integral basis: 
IntegralBasis(K);

// Class group: 
ClassGroup(K);

// Narrow class group: 
NarrowClassGroup(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^24 - 2*x^23 + 119*x^22 - 123*x^21 - 2284*x^20 + 35021*x^19 - 287932*x^18 - 3528451*x^17 + 430787*x^16 + 76267881*x^15 + 383091958*x^14 + 1547361734*x^13 + 1926262502*x^12 - 24636617534*x^11 - 181070920997*x^10 - 710772344063*x^9 - 1879981260494*x^8 - 2702189457572*x^7 + 3316968239109*x^6 + 29845357531527*x^5 + 75565793089194*x^4 + 102052357360777*x^3 + 77691878905336*x^2 + 30942834489368*x + 4794247932259);
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: 
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))];