// Magma code for working with number field 45.45.16231657867656434784079327524611544008919694606149931176692579515382388545323258023038948936636103876569768641.1

// 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 - 4*x^44 - 182*x^43 + 580*x^42 + 15229*x^41 - 35784*x^40 - 771497*x^39 + 1192202*x^38 + 26249095*x^37 - 21282586*x^36 - 630350986*x^35 + 101314575*x^34 + 10957230656*x^33 + 4486413137*x^32 - 139417559409*x^31 - 128053969474*x^30 + 1299055132435*x^29 + 1816090021004*x^28 - 8766783201604*x^27 - 16658946907076*x^26 + 41634175647095*x^25 + 105419181053468*x^24 - 129801264303153*x^23 - 468263377533416*x^22 + 210417608171782*x^21 + 1454214352745320*x^20 + 113648989333176*x^19 - 3097196961798623*x^18 - 1450605309591570*x^17 + 4363534978783893*x^16 + 3500531720706214*x^15 - 3807203164617687*x^14 - 4497701352355764*x^13 + 1760587964112158*x^12 + 3382378542501223*x^11 - 165688879950870*x^10 - 1480388191573288*x^9 - 205996027224400*x^8 + 358811476146228*x^7 + 88160595048203*x^6 - 42989403930482*x^5 - 13014696239433*x^4 + 1879419740489*x^3 + 646702182070*x^2 + 19352782133*x - 2280393187);

// 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 - 4*x^44 - 182*x^43 + 580*x^42 + 15229*x^41 - 35784*x^40 - 771497*x^39 + 1192202*x^38 + 26249095*x^37 - 21282586*x^36 - 630350986*x^35 + 101314575*x^34 + 10957230656*x^33 + 4486413137*x^32 - 139417559409*x^31 - 128053969474*x^30 + 1299055132435*x^29 + 1816090021004*x^28 - 8766783201604*x^27 - 16658946907076*x^26 + 41634175647095*x^25 + 105419181053468*x^24 - 129801264303153*x^23 - 468263377533416*x^22 + 210417608171782*x^21 + 1454214352745320*x^20 + 113648989333176*x^19 - 3097196961798623*x^18 - 1450605309591570*x^17 + 4363534978783893*x^16 + 3500531720706214*x^15 - 3807203164617687*x^14 - 4497701352355764*x^13 + 1760587964112158*x^12 + 3382378542501223*x^11 - 165688879950870*x^10 - 1480388191573288*x^9 - 205996027224400*x^8 + 358811476146228*x^7 + 88160595048203*x^6 - 42989403930482*x^5 - 13014696239433*x^4 + 1879419740489*x^3 + 646702182070*x^2 + 19352782133*x - 2280393187);
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))];