// Magma code for working with number field 42.0.180282079628321418522579756639824623344453525380673158224385384254625263736344592716927.1

// 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 + 2*x^40 + 80*x^39 - 73*x^38 + 139*x^37 + 2635*x^36 - 2181*x^35 + 3934*x^34 + 46702*x^33 - 34779*x^32 + 59013*x^31 + 489577*x^30 - 325107*x^29 + 514966*x^28 + 3148122*x^27 - 1839378*x^26 + 2684924*x^25 + 12529404*x^24 - 6257436*x^23 + 8056643*x^22 + 30745048*x^21 - 12383236*x^20 + 11933159*x^19 + 46384476*x^18 - 13840937*x^17 + 3649186*x^16 + 41025120*x^15 - 8138959*x^14 - 9679906*x^13 + 28379769*x^12 - 8357477*x^11 + 3513515*x^10 + 45545418*x^9 - 12952087*x^8 + 8843411*x^7 + 29858262*x^6 - 8807791*x^5 + 14296232*x^4 - 4154414*x^3 - 181830*x^2 + 2389098*x + 733913);

// 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 + 2*x^40 + 80*x^39 - 73*x^38 + 139*x^37 + 2635*x^36 - 2181*x^35 + 3934*x^34 + 46702*x^33 - 34779*x^32 + 59013*x^31 + 489577*x^30 - 325107*x^29 + 514966*x^28 + 3148122*x^27 - 1839378*x^26 + 2684924*x^25 + 12529404*x^24 - 6257436*x^23 + 8056643*x^22 + 30745048*x^21 - 12383236*x^20 + 11933159*x^19 + 46384476*x^18 - 13840937*x^17 + 3649186*x^16 + 41025120*x^15 - 8138959*x^14 - 9679906*x^13 + 28379769*x^12 - 8357477*x^11 + 3513515*x^10 + 45545418*x^9 - 12952087*x^8 + 8843411*x^7 + 29858262*x^6 - 8807791*x^5 + 14296232*x^4 - 4154414*x^3 - 181830*x^2 + 2389098*x + 733913);
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))];