// Magma code for working with number field 35.35.1455622807785591094953547155658149343464416905925495778881639129313848614446169.1

// (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![859, 136327, 5060011, 70736582, 349101928, -47622511, -4634581071, -7226936109, 16263455858, 42855721796, -10380297650, -94158719636, -38675667681, 92864249975, 76695393961, -39347873390, -57619940147, 3088781711, 23234081997, 3615565971, -5621246939, -1630509423, 850835561, 347984933, -80359740, -44769607, 4452851, 3683986, -112597, -195036, -1020, 6435, 124, -121, -2, 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];