// Magma code for working with number field 27.27.1710104639727978317616739861182036266774010607729523946446049.1

// 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 - 252*x^25 - 189*x^24 + 26676*x^23 + 37278*x^22 - 1542249*x^21 - 2997459*x^20 + 53109180*x^19 + 127595412*x^18 - 1115642646*x^17 - 3116503386*x^16 + 14106344724*x^15 + 44288457588*x^14 - 103097199771*x^13 - 357917273712*x^12 + 414062657892*x^11 + 1585147569417*x^10 - 865139058768*x^9 - 3663417923046*x^8 + 907297669062*x^7 + 4289785218828*x^6 - 435357630576*x^5 - 2278964378115*x^4 + 94666216968*x^3 + 374720442165*x^2 - 29638905030*x - 7668478603);

// 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 - 252*x^25 - 189*x^24 + 26676*x^23 + 37278*x^22 - 1542249*x^21 - 2997459*x^20 + 53109180*x^19 + 127595412*x^18 - 1115642646*x^17 - 3116503386*x^16 + 14106344724*x^15 + 44288457588*x^14 - 103097199771*x^13 - 357917273712*x^12 + 414062657892*x^11 + 1585147569417*x^10 - 865139058768*x^9 - 3663417923046*x^8 + 907297669062*x^7 + 4289785218828*x^6 - 435357630576*x^5 - 2278964378115*x^4 + 94666216968*x^3 + 374720442165*x^2 - 29638905030*x - 7668478603);
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))];