// Magma code for working with number field 42.42.565343212441678035532894502003808167878401992443661947648452445739810658542578516149.1

// 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 - 81*x^40 - 4*x^39 + 2925*x^38 + 276*x^37 - 62221*x^36 - 8388*x^35 + 868038*x^34 + 148164*x^33 - 8369604*x^32 - 1690536*x^31 + 57288192*x^30 + 13101552*x^29 - 281708427*x^28 - 70649728*x^27 + 997233012*x^26 + 267447468*x^25 - 2528669823*x^24 - 709905600*x^23 + 4548805662*x^22 + 1311105276*x^21 - 5738345406*x^20 - 1666490856*x^19 + 5023839240*x^18 + 1442807064*x^17 - 3028872840*x^16 - 846440256*x^15 + 1246931667*x^14 + 334448784*x^13 - 345422658*x^12 - 87695253*x^11 + 62698236*x^10 + 14793089*x^9 - 7127757*x^8 - 1516980*x^7 + 472108*x^6 + 85806*x^5 - 16338*x^4 - 2275*x^3 + 252*x^2 + 21*x - 1);

// 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 - 81*x^40 - 4*x^39 + 2925*x^38 + 276*x^37 - 62221*x^36 - 8388*x^35 + 868038*x^34 + 148164*x^33 - 8369604*x^32 - 1690536*x^31 + 57288192*x^30 + 13101552*x^29 - 281708427*x^28 - 70649728*x^27 + 997233012*x^26 + 267447468*x^25 - 2528669823*x^24 - 709905600*x^23 + 4548805662*x^22 + 1311105276*x^21 - 5738345406*x^20 - 1666490856*x^19 + 5023839240*x^18 + 1442807064*x^17 - 3028872840*x^16 - 846440256*x^15 + 1246931667*x^14 + 334448784*x^13 - 345422658*x^12 - 87695253*x^11 + 62698236*x^10 + 14793089*x^9 - 7127757*x^8 - 1516980*x^7 + 472108*x^6 + 85806*x^5 - 16338*x^4 - 2275*x^3 + 252*x^2 + 21*x - 1);
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))];