// Magma code for working with number field 42.42.8050468075656610214837511220114705524038488445061950919170859146595001220703125.1

// 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 - 63*x^40 + 1764*x^38 - 29043*x^36 - 29*x^35 + 313845*x^34 + 1295*x^33 - 2356263*x^32 - 24605*x^31 + 12710649*x^30 + 262360*x^29 - 50338309*x^28 - 1751015*x^27 + 148468453*x^26 + 7757582*x^25 - 329114310*x^24 - 23675820*x^23 + 550880022*x^22 + 50961633*x^21 - 696364921*x^20 - 78352652*x^19 + 661675518*x^18 + 86284422*x^17 - 467756891*x^16 - 67555180*x^15 + 241883075*x^14 + 36893815*x^13 - 89215119*x^12 - 13584956*x^11 + 22638112*x^10 + 3191321*x^9 - 3752308*x^8 - 437383*x^7 + 374801*x^6 + 29995*x^5 - 19355*x^4 - 798*x^3 + 343*x^2 + 14*x - 1);

// 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 - 63*x^40 + 1764*x^38 - 29043*x^36 - 29*x^35 + 313845*x^34 + 1295*x^33 - 2356263*x^32 - 24605*x^31 + 12710649*x^30 + 262360*x^29 - 50338309*x^28 - 1751015*x^27 + 148468453*x^26 + 7757582*x^25 - 329114310*x^24 - 23675820*x^23 + 550880022*x^22 + 50961633*x^21 - 696364921*x^20 - 78352652*x^19 + 661675518*x^18 + 86284422*x^17 - 467756891*x^16 - 67555180*x^15 + 241883075*x^14 + 36893815*x^13 - 89215119*x^12 - 13584956*x^11 + 22638112*x^10 + 3191321*x^9 - 3752308*x^8 - 437383*x^7 + 374801*x^6 + 29995*x^5 - 19355*x^4 - 798*x^3 + 343*x^2 + 14*x - 1);
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))];