// Magma code for working with number field 42.42.776706980099412683270456983716625307007021218630434418395046564476510573119136503693312.1

// 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 - 105*x^40 + 4977*x^38 - 140819*x^36 - 2*x^35 + 2650095*x^34 + 28*x^33 - 34991649*x^32 + 3416*x^31 + 333300632*x^30 - 147910*x^29 - 2319579309*x^28 + 2698850*x^27 + 11820066804*x^26 - 26976474*x^25 - 43812582639*x^24 + 153194076*x^23 + 116368453593*x^22 - 445815310*x^21 - 216225058245*x^20 + 272316800*x^19 + 271556250546*x^18 + 1986860386*x^17 - 220004406804*x^16 - 5445985720*x^15 + 108239023559*x^14 + 5365080098*x^13 - 29954210552*x^12 - 2176838734*x^11 + 4303639312*x^10 + 340937478*x^9 - 323375283*x^8 - 25471098*x^7 + 12576760*x^6 + 955878*x^5 - 232127*x^4 - 16898*x^3 + 1505*x^2 + 112*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 - 105*x^40 + 4977*x^38 - 140819*x^36 - 2*x^35 + 2650095*x^34 + 28*x^33 - 34991649*x^32 + 3416*x^31 + 333300632*x^30 - 147910*x^29 - 2319579309*x^28 + 2698850*x^27 + 11820066804*x^26 - 26976474*x^25 - 43812582639*x^24 + 153194076*x^23 + 116368453593*x^22 - 445815310*x^21 - 216225058245*x^20 + 272316800*x^19 + 271556250546*x^18 + 1986860386*x^17 - 220004406804*x^16 - 5445985720*x^15 + 108239023559*x^14 + 5365080098*x^13 - 29954210552*x^12 - 2176838734*x^11 + 4303639312*x^10 + 340937478*x^9 - 323375283*x^8 - 25471098*x^7 + 12576760*x^6 + 955878*x^5 - 232127*x^4 - 16898*x^3 + 1505*x^2 + 112*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))];