// Magma code for working with number field 45.45.202328376069000568231882298692586415768387771371083469740034658547467305567124640219844877719879150390625.1

// 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^45 - 165*x^43 - 25*x^42 + 12060*x^41 + 3486*x^40 - 519330*x^39 - 214140*x^38 + 14776560*x^37 + 7718745*x^36 - 295077669*x^35 - 182852610*x^34 + 4284179615*x^33 + 3016157820*x^32 - 46197393465*x^31 - 35772719505*x^30 + 374607222195*x^29 + 310484223975*x^28 - 2298730688540*x^27 - 1988244775980*x^26 + 10695933805596*x^25 + 9407317026635*x^24 - 37696094037390*x^23 - 32764647222885*x^22 + 100289481110340*x^21 + 83332538318277*x^20 - 200279250507195*x^19 - 152889355168310*x^18 + 297466429398705*x^17 + 198745289480940*x^16 - 323624520119411*x^15 - 178210655500665*x^14 + 251547979741875*x^13 + 105785393431525*x^12 - 134218212107550*x^11 - 38927292179565*x^10 + 46104688292390*x^9 + 7956837983055*x^8 - 9173261413650*x^7 - 765305596255*x^6 + 891053703312*x^5 + 40709257785*x^4 - 38148076985*x^3 - 1973427960*x^2 + 606052050*x + 47691757);

// 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^45 - 165*x^43 - 25*x^42 + 12060*x^41 + 3486*x^40 - 519330*x^39 - 214140*x^38 + 14776560*x^37 + 7718745*x^36 - 295077669*x^35 - 182852610*x^34 + 4284179615*x^33 + 3016157820*x^32 - 46197393465*x^31 - 35772719505*x^30 + 374607222195*x^29 + 310484223975*x^28 - 2298730688540*x^27 - 1988244775980*x^26 + 10695933805596*x^25 + 9407317026635*x^24 - 37696094037390*x^23 - 32764647222885*x^22 + 100289481110340*x^21 + 83332538318277*x^20 - 200279250507195*x^19 - 152889355168310*x^18 + 297466429398705*x^17 + 198745289480940*x^16 - 323624520119411*x^15 - 178210655500665*x^14 + 251547979741875*x^13 + 105785393431525*x^12 - 134218212107550*x^11 - 38927292179565*x^10 + 46104688292390*x^9 + 7956837983055*x^8 - 9173261413650*x^7 - 765305596255*x^6 + 891053703312*x^5 + 40709257785*x^4 - 38148076985*x^3 - 1973427960*x^2 + 606052050*x + 47691757);
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))];