// Magma code for working with number field 35.35.10607266494966666158512469468409065269845427817856998775062284008873509080731241.1

// (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![-27756643, 172726939, 146105313, -2696920777, 3432884529, 11871839347, -32473918206, -451900172, 86642725878, -87564669300, -60624002242, 165504701751, -62920019542, -96083394992, 100720726682, 736744022, -47903559470, 19862461891, 8653560326, -8565464505, 444168124, 1664362308, -471756827, -154446290, 89830596, 2494886, -8797227, 885684, 492148, -93149, -15350, 4336, 231, -102, -1, 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];