// Magma code for working with number field 45.45.6444338306249279600681470320699578378786756892621858688031629726344906572421677992679178714752197265625.1

// 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 - 135*x^43 - 45*x^42 + 8145*x^41 + 5076*x^40 - 290700*x^39 - 253620*x^38 + 6857235*x^37 + 7452610*x^36 - 113268978*x^35 - 144333495*x^34 + 1355213265*x^33 + 1957216455*x^32 - 11983335705*x^31 - 19266777171*x^30 + 79141746870*x^29 + 140783463765*x^28 - 391240008255*x^27 - 773615354895*x^26 + 1437267934764*x^25 + 3215872493700*x^24 - 3839971488165*x^23 - 10109340177300*x^22 + 7085017431465*x^21 + 23873005106874*x^20 - 7751347674075*x^19 - 41764000577760*x^18 + 1270029598110*x^17 + 52868417349600*x^16 + 10969424061639*x^15 - 46623586775670*x^14 - 19170786833010*x^13 + 26882974463550*x^12 + 16378966497285*x^11 - 8981328155022*x^10 - 7825331392520*x^9 + 1236586926825*x^8 + 1998991148400*x^7 + 84870189090*x^6 - 236319447555*x^5 - 31508573040*x^4 + 10090496085*x^3 + 1414058625*x^2 - 109748520*x - 10224199);

// 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 - 135*x^43 - 45*x^42 + 8145*x^41 + 5076*x^40 - 290700*x^39 - 253620*x^38 + 6857235*x^37 + 7452610*x^36 - 113268978*x^35 - 144333495*x^34 + 1355213265*x^33 + 1957216455*x^32 - 11983335705*x^31 - 19266777171*x^30 + 79141746870*x^29 + 140783463765*x^28 - 391240008255*x^27 - 773615354895*x^26 + 1437267934764*x^25 + 3215872493700*x^24 - 3839971488165*x^23 - 10109340177300*x^22 + 7085017431465*x^21 + 23873005106874*x^20 - 7751347674075*x^19 - 41764000577760*x^18 + 1270029598110*x^17 + 52868417349600*x^16 + 10969424061639*x^15 - 46623586775670*x^14 - 19170786833010*x^13 + 26882974463550*x^12 + 16378966497285*x^11 - 8981328155022*x^10 - 7825331392520*x^9 + 1236586926825*x^8 + 1998991148400*x^7 + 84870189090*x^6 - 236319447555*x^5 - 31508573040*x^4 + 10090496085*x^3 + 1414058625*x^2 - 109748520*x - 10224199);
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))];