// Magma code for working with number field 45.45.16527441490674381067348948889146522048718287580785548225593089356497792068001085725808918264262962961.1

// (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![-36851, 2118031, -40977465, 298954277, -166352258, -6916885901, 16257686900, 64595300484, -182597145321, -333732759991, 957147248584, 1039444940408, -2929193071000, -2048306537737, 5804631677810, 2645087732416, -7914078560767, -2271295840968, 7719490702430, 1265687493078, -5529580093472, -394556499523, 2959992694385, 511532340, -1197228116258, 66266177672, 368042106614, -36476021474, -86102308596, 11523689449, 15280860430, -2468141099, -2040701711, 374739344, 202181480, -40702810, -14526130, 3130045, 729869, -165566, -24132, 5696, 468, -114, -4, 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];