// Magma code for working with number field 38.0.2472960613492762938009352687218362626942035203162587025151809700624254848124906369677396881178624.1

// (Note that not all these functions may be available, and some may take a long time to execute.)

// Define the number field: 
R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![63175314409, 0, 1811471395871, 0, 20190511272319, 0, 111675903836641, 0, 333058518278260, 0, 575420256350720, 0, 609555286857416, 0, 413962321806804, 0, 187171442808726, 0, 58044125409037, 0, 12621446315235, 0, 1954462763213, 0, 217550145613, 0, 17454519530, 0, 1004488522, 0, 40850264, 0, 1139565, 0, 20630, 0, 217, 0, 1]);

// Defining polynomial: 
DefiningPolynomial(K);

// Degree over Q: 
Degree(K);

// Signature: 
Signature(K);

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

// Ramified primes: 
PrimeDivisors(Discriminant(Integers(K)));

// Integral basis: 
IntegralBasis(K);

// Class group: 
ClassGroup(K);

// Unit group: 
UK, f := UnitGroup(K);

// Unit rank: 
UnitRank(K);

// Generator for roots of unity: 
K!f(TU.1) where TU,f is TorsionUnitGroup(K);

// Fundamental units: 
[K!f(g): g in Generators(UK)];

// Regulator: 
Regulator(K);

// Galois group: 
GaloisGroup(K);

// Frobenius cycle types: 
p := 7;  // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
idealfactors := Factorization(p*Integers(K));  // get the data
[<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];