// Magma code for working with number field 42.0.124938828493629533000907736270182756286928467256210015044892719712262808512338377882811.1

// 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 - 21*x^41 + 231*x^40 - 1750*x^39 + 10395*x^38 - 52269*x^37 + 233611*x^36 - 952918*x^35 + 3599785*x^34 - 12710201*x^33 + 42287014*x^32 - 133370664*x^31 + 400618099*x^30 - 1149584975*x^29 + 3160451592*x^28 - 8341179427*x^27 + 21174580693*x^26 - 51755274760*x^25 + 121953514697*x^24 - 277155797093*x^23 + 608007380806*x^22 - 1287295716241*x^21 + 2631847521996*x^20 - 5192066875520*x^19 + 9886990587068*x^18 - 18146977046303*x^17 + 32112635555588*x^16 - 54655115299659*x^15 + 89504829240548*x^14 - 140479267770793*x^13 + 211514505374151*x^12 - 303526580264439*x^11 + 416119886096880*x^10 - 538840804150264*x^9 + 662937764461147*x^8 - 758558920220686*x^7 + 819087800747125*x^6 - 798550011094073*x^5 + 729734808064227*x^4 - 562691756062212*x^3 + 407486767661645*x^2 - 201294019183845*x + 99519182315771);

// 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 - 21*x^41 + 231*x^40 - 1750*x^39 + 10395*x^38 - 52269*x^37 + 233611*x^36 - 952918*x^35 + 3599785*x^34 - 12710201*x^33 + 42287014*x^32 - 133370664*x^31 + 400618099*x^30 - 1149584975*x^29 + 3160451592*x^28 - 8341179427*x^27 + 21174580693*x^26 - 51755274760*x^25 + 121953514697*x^24 - 277155797093*x^23 + 608007380806*x^22 - 1287295716241*x^21 + 2631847521996*x^20 - 5192066875520*x^19 + 9886990587068*x^18 - 18146977046303*x^17 + 32112635555588*x^16 - 54655115299659*x^15 + 89504829240548*x^14 - 140479267770793*x^13 + 211514505374151*x^12 - 303526580264439*x^11 + 416119886096880*x^10 - 538840804150264*x^9 + 662937764461147*x^8 - 758558920220686*x^7 + 819087800747125*x^6 - 798550011094073*x^5 + 729734808064227*x^4 - 562691756062212*x^3 + 407486767661645*x^2 - 201294019183845*x + 99519182315771);
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))];