// Magma code for working with number field 40.0.34296436716006088086348892675736212426840072977067776247113342729404166276203830051422119140625.1.
// 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^40 - x^39 + 205*x^38 - 210*x^37 + 17179*x^36 - 18233*x^35 + 769166*x^34 - 861175*x^33 + 20231634*x^32 - 24611285*x^31 + 321376610*x^30 - 443994502*x^29 + 2934250927*x^28 - 5127881369*x^27 + 11052709879*x^26 - 38235186403*x^25 - 47987734851*x^24 - 204754941776*x^23 - 587996815784*x^22 - 1156468185533*x^21 + 697453826091*x^20 - 6137453171112*x^19 + 39496054736893*x^18 + 2277515781046*x^17 + 279072370692972*x^16 + 222049495807135*x^15 + 1191232934278609*x^14 + 327805687794932*x^13 + 5697446694340888*x^12 - 4841562954273985*x^11 + 31699026105567100*x^10 - 7800767151617665*x^9 + 141832220187432461*x^8 + 82933946634742001*x^7 + 479398723525712355*x^6 + 288175051685606927*x^5 + 1062431436014171077*x^4 + 355626436037277912*x^3 + 1157271788184170842*x^2 + 252128673371738912*x + 553402136113967801);

// Defining polynomial: 
DefiningPolynomial(K);

// Degree over Q: 
Degree(K);

// Signature: 
Signature(K);

// Discriminant: 
OK := Integers(K); Discriminant(OK);

// Ramified primes: 
PrimeDivisors(Discriminant(OK));

// Automorphisms: 
Automorphisms(K);

// Integral basis: 
IntegralBasis(K);

// Class group: 
ClassGroup(K);

// Narrow class group: 
NarrowClassGroup(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(Rationals()); K<a> := NumberField(x^40 - x^39 + 205*x^38 - 210*x^37 + 17179*x^36 - 18233*x^35 + 769166*x^34 - 861175*x^33 + 20231634*x^32 - 24611285*x^31 + 321376610*x^30 - 443994502*x^29 + 2934250927*x^28 - 5127881369*x^27 + 11052709879*x^26 - 38235186403*x^25 - 47987734851*x^24 - 204754941776*x^23 - 587996815784*x^22 - 1156468185533*x^21 + 697453826091*x^20 - 6137453171112*x^19 + 39496054736893*x^18 + 2277515781046*x^17 + 279072370692972*x^16 + 222049495807135*x^15 + 1191232934278609*x^14 + 327805687794932*x^13 + 5697446694340888*x^12 - 4841562954273985*x^11 + 31699026105567100*x^10 - 7800767151617665*x^9 + 141832220187432461*x^8 + 82933946634742001*x^7 + 479398723525712355*x^6 + 288175051685606927*x^5 + 1062431436014171077*x^4 + 355626436037277912*x^3 + 1157271788184170842*x^2 + 252128673371738912*x + 553402136113967801);
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 pO_K for p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];