// Magma code for working with number field 38.38.687661045808093482376579225097085116287670624782479569073265850359469722789446885178751177457664.1

// (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![-1101076991, 0, 744125807285, 0, -7629511512714, 0, 31469275400290, 0, -71207905317537, 0, 100459885220950, 0, -94680464008195, 0, 62167403988174, 0, -29202012585177, 0, 9974553348782, 0, -2499589479956, 0, 460944322269, 0, -62411853712, 0, 6157331558, 0, -436357836, 0, 21705622, 0, -730575, 0, 15662, 0, -191, 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];