// Magma code for working with number field 42.0.2281836760183646137444154412268560109828024514076489472840222217265158917203.1

// 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 - x^41 + 21*x^40 - 18*x^39 + 248*x^38 - 191*x^37 + 1996*x^36 - 1375*x^35 + 12088*x^34 - 7495*x^33 + 57373*x^32 - 31825*x^31 + 219409*x^30 - 108700*x^29 + 684645*x^28 - 300153*x^27 + 1757705*x^26 - 677756*x^25 + 3715466*x^24 - 1242500*x^23 + 6455978*x^22 - 1853597*x^21 + 9148985*x^20 - 2205194*x^19 + 10469894*x^18 - 2090336*x^17 + 9505112*x^16 - 1507895*x^15 + 6709547*x^14 - 839645*x^13 + 3559478*x^12 - 314951*x^11 + 1368367*x^10 - 93808*x^9 + 355278*x^8 - 10362*x^7 + 58036*x^6 - 2497*x^5 + 4950*x^4 + 165*x^3 + 176*x^2 - 11*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 - x^41 + 21*x^40 - 18*x^39 + 248*x^38 - 191*x^37 + 1996*x^36 - 1375*x^35 + 12088*x^34 - 7495*x^33 + 57373*x^32 - 31825*x^31 + 219409*x^30 - 108700*x^29 + 684645*x^28 - 300153*x^27 + 1757705*x^26 - 677756*x^25 + 3715466*x^24 - 1242500*x^23 + 6455978*x^22 - 1853597*x^21 + 9148985*x^20 - 2205194*x^19 + 10469894*x^18 - 2090336*x^17 + 9505112*x^16 - 1507895*x^15 + 6709547*x^14 - 839645*x^13 + 3559478*x^12 - 314951*x^11 + 1368367*x^10 - 93808*x^9 + 355278*x^8 - 10362*x^7 + 58036*x^6 - 2497*x^5 + 4950*x^4 + 165*x^3 + 176*x^2 - 11*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))];