// Magma code for working with number field 42.0.2333538056680443170216809877092138324063458978840788486681774071870758537049156298101507.1

// 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 - 11*x^41 + 31*x^40 + 15*x^39 + 149*x^38 - 2171*x^37 + 5622*x^36 - 5813*x^35 + 17703*x^34 - 77949*x^33 + 215171*x^32 - 500434*x^31 + 1078714*x^30 - 3010069*x^29 + 7913255*x^28 - 11776682*x^27 + 16679313*x^26 - 87643658*x^25 + 355007978*x^24 - 761687128*x^23 + 908129906*x^22 - 907038305*x^21 + 2012009714*x^20 - 6104907919*x^19 + 18723809190*x^18 - 48743446573*x^17 + 87856746678*x^16 - 105007899193*x^15 + 97597592982*x^14 - 99989052476*x^13 + 101465465536*x^12 - 67260548547*x^11 + 47339909902*x^10 - 82545626769*x^9 + 101683114718*x^8 - 83757841269*x^7 + 103258810875*x^6 - 130872100909*x^5 + 93586325459*x^4 - 47886183228*x^3 + 41077962943*x^2 - 30874848118*x + 9042834947);

// 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 - 11*x^41 + 31*x^40 + 15*x^39 + 149*x^38 - 2171*x^37 + 5622*x^36 - 5813*x^35 + 17703*x^34 - 77949*x^33 + 215171*x^32 - 500434*x^31 + 1078714*x^30 - 3010069*x^29 + 7913255*x^28 - 11776682*x^27 + 16679313*x^26 - 87643658*x^25 + 355007978*x^24 - 761687128*x^23 + 908129906*x^22 - 907038305*x^21 + 2012009714*x^20 - 6104907919*x^19 + 18723809190*x^18 - 48743446573*x^17 + 87856746678*x^16 - 105007899193*x^15 + 97597592982*x^14 - 99989052476*x^13 + 101465465536*x^12 - 67260548547*x^11 + 47339909902*x^10 - 82545626769*x^9 + 101683114718*x^8 - 83757841269*x^7 + 103258810875*x^6 - 130872100909*x^5 + 93586325459*x^4 - 47886183228*x^3 + 41077962943*x^2 - 30874848118*x + 9042834947);
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))];