// Magma code for working with number field 42.0.56353276529596271503862578540802938668269419115433656434196014026165008544921875.1

// 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 + 42*x^40 + 819*x^38 + 9842*x^36 - 29*x^35 + 81585*x^34 - 1015*x^33 + 494802*x^32 - 16240*x^31 + 2272424*x^30 - 157325*x^29 + 8070266*x^28 - 1030225*x^27 + 22451828*x^26 - 4821453*x^25 + 49380100*x^24 - 16625700*x^23 + 86841307*x^22 - 42945897*x^21 + 125155604*x^20 - 83971762*x^19 + 155582203*x^18 - 126598108*x^17 + 178243674*x^16 - 155985200*x^15 + 191428385*x^14 - 177440270*x^13 + 185472266*x^12 - 199822441*x^11 + 177861992*x^10 - 194249279*x^9 + 208189352*x^8 - 152082148*x^7 + 226081541*x^6 - 195873685*x^5 + 141819720*x^4 - 297986948*x^3 + 34945918*x^2 - 144775946*x + 599786069);

// 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 + 42*x^40 + 819*x^38 + 9842*x^36 - 29*x^35 + 81585*x^34 - 1015*x^33 + 494802*x^32 - 16240*x^31 + 2272424*x^30 - 157325*x^29 + 8070266*x^28 - 1030225*x^27 + 22451828*x^26 - 4821453*x^25 + 49380100*x^24 - 16625700*x^23 + 86841307*x^22 - 42945897*x^21 + 125155604*x^20 - 83971762*x^19 + 155582203*x^18 - 126598108*x^17 + 178243674*x^16 - 155985200*x^15 + 191428385*x^14 - 177440270*x^13 + 185472266*x^12 - 199822441*x^11 + 177861992*x^10 - 194249279*x^9 + 208189352*x^8 - 152082148*x^7 + 226081541*x^6 - 195873685*x^5 + 141819720*x^4 - 297986948*x^3 + 34945918*x^2 - 144775946*x + 599786069);
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))];