// Magma code for working with number field 42.0.213117637842219136349783160187126852219651508028407629359584264706447783442638664863363.1

// 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 - 4*x^41 + 62*x^40 - 200*x^39 + 2132*x^38 - 6208*x^37 + 46584*x^36 - 115568*x^35 + 701352*x^34 - 1537594*x^33 + 7769821*x^32 - 14875116*x^31 + 64656220*x^30 - 109927490*x^29 + 414171141*x^28 - 618138014*x^27 + 2041345075*x^26 - 2683015728*x^25 + 7816607585*x^24 - 8902561137*x^23 + 22976431713*x^22 - 22760113866*x^21 + 52536783259*x^20 - 45434427911*x^19 + 92226880868*x^18 - 71336683352*x^17 + 125874997329*x^16 - 89801104142*x^15 + 131144820279*x^14 - 89289000509*x^13 + 106600416700*x^12 - 67935846970*x^11 + 65774842016*x^10 - 38731708226*x^9 + 30551737937*x^8 - 15767508635*x^7 + 9878705051*x^6 - 4297296085*x^5 + 2147704850*x^4 - 731044046*x^3 + 251922248*x^2 - 47762400*x + 8082649);

// 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 - 4*x^41 + 62*x^40 - 200*x^39 + 2132*x^38 - 6208*x^37 + 46584*x^36 - 115568*x^35 + 701352*x^34 - 1537594*x^33 + 7769821*x^32 - 14875116*x^31 + 64656220*x^30 - 109927490*x^29 + 414171141*x^28 - 618138014*x^27 + 2041345075*x^26 - 2683015728*x^25 + 7816607585*x^24 - 8902561137*x^23 + 22976431713*x^22 - 22760113866*x^21 + 52536783259*x^20 - 45434427911*x^19 + 92226880868*x^18 - 71336683352*x^17 + 125874997329*x^16 - 89801104142*x^15 + 131144820279*x^14 - 89289000509*x^13 + 106600416700*x^12 - 67935846970*x^11 + 65774842016*x^10 - 38731708226*x^9 + 30551737937*x^8 - 15767508635*x^7 + 9878705051*x^6 - 4297296085*x^5 + 2147704850*x^4 - 731044046*x^3 + 251922248*x^2 - 47762400*x + 8082649);
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))];