// Magma code for working with number field 20.20.1186334452527276032000000000000000.1.
// 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^20 - 6*x^19 - 30*x^18 + 212*x^17 + 296*x^16 - 2868*x^15 - 704*x^14 + 18976*x^13 - 6340*x^12 - 64384*x^11 + 46636*x^10 + 103920*x^9 - 114392*x^8 - 52928*x^7 + 104592*x^6 - 22864*x^5 - 15984*x^4 + 5632*x^3 + 512*x^2 - 288*x + 16);

// Defining polynomial: 
DefiningPolynomial(K);

// Degree over Q: 
Degree(K);

// Signature: 
Signature(K);

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

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

// Automorphisms: 
Automorphisms(K);

// Integral basis: 
IntegralBasis(K);

// Class group: 
ClassGroup(K);

// Narrow class group: 
NarrowClassGroup(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(Rationals()); K<a> := NumberField(x^20 - 6*x^19 - 30*x^18 + 212*x^17 + 296*x^16 - 2868*x^15 - 704*x^14 + 18976*x^13 - 6340*x^12 - 64384*x^11 + 46636*x^10 + 103920*x^9 - 114392*x^8 - 52928*x^7 + 104592*x^6 - 22864*x^5 - 15984*x^4 + 5632*x^3 + 512*x^2 - 288*x + 16);
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 pO_K for p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];