// Magma code for working with number field 45.45.112369590570788381690576561372238600351878236661366984431016423673684387952713968402481343001103760716137281.1

// 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 - x^44 - 132*x^43 + 301*x^42 + 7581*x^41 - 27065*x^40 - 236809*x^39 + 1229173*x^38 + 3960874*x^37 - 33065607*x^36 - 18797995*x^35 + 555716542*x^34 - 619527603*x^33 - 5732299123*x^32 + 15235581842*x^31 + 30609318626*x^30 - 168897578817*x^29 + 17112929652*x^28 + 1054352970448*x^27 - 1501837759883*x^26 - 3320354588485*x^25 + 10782132568732*x^24 - 107051646226*x^23 - 36936809485370*x^22 + 41830258905956*x^21 + 53083505529011*x^20 - 152445251342987*x^19 + 42842896041923*x^18 + 229594603485553*x^17 - 272222797945867*x^16 - 70739760279828*x^15 + 364126674087072*x^14 - 209159107972130*x^13 - 142016333800362*x^12 + 231101895832035*x^11 - 63870399263881*x^10 - 62082302112786*x^9 + 49149720219336*x^8 - 3670701881262*x^7 - 8040338248733*x^6 + 2565873490427*x^5 + 306896523050*x^4 - 230447557112*x^3 + 11171763392*x^2 + 5932497375*x - 637239997);

// 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 - x^44 - 132*x^43 + 301*x^42 + 7581*x^41 - 27065*x^40 - 236809*x^39 + 1229173*x^38 + 3960874*x^37 - 33065607*x^36 - 18797995*x^35 + 555716542*x^34 - 619527603*x^33 - 5732299123*x^32 + 15235581842*x^31 + 30609318626*x^30 - 168897578817*x^29 + 17112929652*x^28 + 1054352970448*x^27 - 1501837759883*x^26 - 3320354588485*x^25 + 10782132568732*x^24 - 107051646226*x^23 - 36936809485370*x^22 + 41830258905956*x^21 + 53083505529011*x^20 - 152445251342987*x^19 + 42842896041923*x^18 + 229594603485553*x^17 - 272222797945867*x^16 - 70739760279828*x^15 + 364126674087072*x^14 - 209159107972130*x^13 - 142016333800362*x^12 + 231101895832035*x^11 - 63870399263881*x^10 - 62082302112786*x^9 + 49149720219336*x^8 - 3670701881262*x^7 - 8040338248733*x^6 + 2565873490427*x^5 + 306896523050*x^4 - 230447557112*x^3 + 11171763392*x^2 + 5932497375*x - 637239997);
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))];