// Magma code for working with number field 42.0.409216671487706896433400972530400813921910324685198934110144125610159603082437813389667.1

// 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 - x^41 + 20*x^40 - 25*x^39 - 8*x^38 - 138*x^37 - 2077*x^36 + 839*x^35 - 7305*x^34 + 8805*x^33 + 67596*x^32 - 12107*x^31 + 444232*x^30 - 141646*x^29 + 238560*x^28 + 2150502*x^27 - 3285732*x^26 + 14361861*x^25 - 708679*x^24 + 4004385*x^23 + 63920534*x^22 - 73366557*x^21 + 232544983*x^20 + 114300646*x^19 + 134380032*x^18 + 644451241*x^17 - 110366019*x^16 + 50739485*x^15 + 3000292231*x^14 - 3874326960*x^13 + 7531027885*x^12 - 11845394651*x^11 + 8115584321*x^10 - 10398559445*x^9 + 11442787794*x^8 - 3901741530*x^7 + 3575301015*x^6 - 3614499453*x^5 - 715331685*x^4 + 331663193*x^3 + 2507361126*x^2 - 1452650295*x + 649728353);

// 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 - x^41 + 20*x^40 - 25*x^39 - 8*x^38 - 138*x^37 - 2077*x^36 + 839*x^35 - 7305*x^34 + 8805*x^33 + 67596*x^32 - 12107*x^31 + 444232*x^30 - 141646*x^29 + 238560*x^28 + 2150502*x^27 - 3285732*x^26 + 14361861*x^25 - 708679*x^24 + 4004385*x^23 + 63920534*x^22 - 73366557*x^21 + 232544983*x^20 + 114300646*x^19 + 134380032*x^18 + 644451241*x^17 - 110366019*x^16 + 50739485*x^15 + 3000292231*x^14 - 3874326960*x^13 + 7531027885*x^12 - 11845394651*x^11 + 8115584321*x^10 - 10398559445*x^9 + 11442787794*x^8 - 3901741530*x^7 + 3575301015*x^6 - 3614499453*x^5 - 715331685*x^4 + 331663193*x^3 + 2507361126*x^2 - 1452650295*x + 649728353);
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))];