// Magma code for working with number field 27.9.17717054310925604811052271173453197606912.2

// Some of these functions may take a long time to execute (this depends on the field).

// Define the number field: 
R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 9*x^26 + 36*x^25 - 54*x^24 - 180*x^23 + 1296*x^22 - 3699*x^21 + 5184*x^20 + 1737*x^19 - 28278*x^18 + 81630*x^17 - 152073*x^16 + 206037*x^15 - 206640*x^14 + 143748*x^13 - 36954*x^12 - 70929*x^11 + 125325*x^10 - 106329*x^9 + 53154*x^8 - 10548*x^7 - 9594*x^6 + 13185*x^5 - 9180*x^4 + 4068*x^3 - 1053*x^2 - 207*x + 163);

// Defining polynomial: 
DefiningPolynomial(K);

// Degree over Q: 
Degree(K);

// Signature: 
Signature(K);

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

// Ramified primes: 
PrimeDivisors(Discriminant(OK));

// Autmorphisms: 
Automorphisms(K);

// Integral basis: 
IntegralBasis(K);

// Class group: 
ClassGroup(K);

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

// Unit rank: 
UnitRank(K);

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

// Fundamental units: 
[K|fUK(g): g in Generators(UK)];

// Regulator: 
Regulator(K);

// Analytic class number formula: 
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^27 - 9*x^26 + 36*x^25 - 54*x^24 - 180*x^23 + 1296*x^22 - 3699*x^21 + 5184*x^20 + 1737*x^19 - 28278*x^18 + 81630*x^17 - 152073*x^16 + 206037*x^15 - 206640*x^14 + 143748*x^13 - 36954*x^12 - 70929*x^11 + 125325*x^10 - 106329*x^9 + 53154*x^8 - 10548*x^7 - 9594*x^6 + 13185*x^5 - 9180*x^4 + 4068*x^3 - 1053*x^2 - 207*x + 163);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

// Intermediate fields: 
L := Subfields(K); L[2..#L];

// Galois group: 
G = GaloisGroup(K);

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