// Magma code for working with number field 42.42.41254041074227118944013302420247071185885898029927512658248841159725538158313472.1

// 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 - 43*x^40 + 860*x^38 - 10621*x^36 + 90687*x^34 - 567987*x^32 + 2701776*x^30 - 9970840*x^28 + 28915436*x^26 - 66335412*x^24 + 120609840*x^22 - 173376645*x^20 + 195747825*x^18 - 171655785*x^16 + 115000920*x^14 - 57500460*x^12 + 20764055*x^10 - 5167525*x^8 + 826804*x^6 - 76153*x^4 + 3311*x^2 - 43);

// 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 - 43*x^40 + 860*x^38 - 10621*x^36 + 90687*x^34 - 567987*x^32 + 2701776*x^30 - 9970840*x^28 + 28915436*x^26 - 66335412*x^24 + 120609840*x^22 - 173376645*x^20 + 195747825*x^18 - 171655785*x^16 + 115000920*x^14 - 57500460*x^12 + 20764055*x^10 - 5167525*x^8 + 826804*x^6 - 76153*x^4 + 3311*x^2 - 43);
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))];