// Magma code for working with number field 42.42.1090030896264192289800449659845679818091197961133776603876122561317234873686091104256.1

// 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^42 - 84*x^40 + 3276*x^38 - 78736*x^36 + 1305360*x^34 - 15833664*x^32 + 145435136*x^30 - 1032886400*x^28 + 5741639680*x^26 - 25131904000*x^24 + 86707088384*x^22 - 234966480896*x^20 + 496154537984*x^18 - 805934137344*x^16 + 988445753344*x^14 - 891877195776*x^12 + 571258175488*x^10 - 247113187328*x^8 + 67166797824*x^6 - 10250354688*x^4 + 719323136*x^2 - 14680064);

// 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^42 - 84*x^40 + 3276*x^38 - 78736*x^36 + 1305360*x^34 - 15833664*x^32 + 145435136*x^30 - 1032886400*x^28 + 5741639680*x^26 - 25131904000*x^24 + 86707088384*x^22 - 234966480896*x^20 + 496154537984*x^18 - 805934137344*x^16 + 988445753344*x^14 - 891877195776*x^12 + 571258175488*x^10 - 247113187328*x^8 + 67166797824*x^6 - 10250354688*x^4 + 719323136*x^2 - 14680064);
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))];