// Magma code for working with number field 27.1.119615770666944050013402329147269064161944026933311.1

// 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 + 56*x^25 - 273*x^24 + 1290*x^23 - 5604*x^22 + 24003*x^21 - 98346*x^20 + 381168*x^19 - 1383408*x^18 + 4700958*x^17 - 15265944*x^16 + 47759673*x^15 - 142609974*x^14 + 403676163*x^13 - 1071234237*x^12 + 2616260829*x^11 - 5766916275*x^10 + 11330239941*x^9 - 19648861461*x^8 + 29452821249*x^7 - 36679431165*x^6 + 35809916697*x^5 - 25389897726*x^4 + 11811942110*x^3 - 3328128234*x^2 + 1063555192*x - 565863243);

// 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 + 56*x^25 - 273*x^24 + 1290*x^23 - 5604*x^22 + 24003*x^21 - 98346*x^20 + 381168*x^19 - 1383408*x^18 + 4700958*x^17 - 15265944*x^16 + 47759673*x^15 - 142609974*x^14 + 403676163*x^13 - 1071234237*x^12 + 2616260829*x^11 - 5766916275*x^10 + 11330239941*x^9 - 19648861461*x^8 + 29452821249*x^7 - 36679431165*x^6 + 35809916697*x^5 - 25389897726*x^4 + 11811942110*x^3 - 3328128234*x^2 + 1063555192*x - 565863243);
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))];