// Magma code for working with number field 45.45.43679806300610465846484971330073185012597520004657724600953543350870941304329239684756561.1

// 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 - 76*x^43 + 71*x^42 + 2587*x^41 - 2253*x^40 - 52277*x^39 + 42329*x^38 + 700945*x^37 - 526048*x^36 - 6604336*x^35 + 4578520*x^34 + 45186997*x^33 - 28840788*x^32 - 228874394*x^31 + 134081171*x^30 + 867295136*x^29 - 465239246*x^28 - 2469432362*x^27 + 1211308198*x^26 + 5279515898*x^25 - 2368332022*x^24 - 8432662207*x^23 + 3466272920*x^22 + 9971146414*x^21 - 3771201953*x^20 - 8615444602*x^19 + 3016692905*x^18 + 5349176614*x^17 - 1747589185*x^16 - 2338480852*x^15 + 719039461*x^14 + 702585559*x^13 - 204971923*x^12 - 140848481*x^11 + 39126498*x^10 + 18138585*x^9 - 4764404*x^8 - 1426972*x^7 + 345013*x^6 + 64118*x^5 - 13350*x^4 - 1490*x^3 + 225*x^2 + 15*x - 1);

// 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 - 76*x^43 + 71*x^42 + 2587*x^41 - 2253*x^40 - 52277*x^39 + 42329*x^38 + 700945*x^37 - 526048*x^36 - 6604336*x^35 + 4578520*x^34 + 45186997*x^33 - 28840788*x^32 - 228874394*x^31 + 134081171*x^30 + 867295136*x^29 - 465239246*x^28 - 2469432362*x^27 + 1211308198*x^26 + 5279515898*x^25 - 2368332022*x^24 - 8432662207*x^23 + 3466272920*x^22 + 9971146414*x^21 - 3771201953*x^20 - 8615444602*x^19 + 3016692905*x^18 + 5349176614*x^17 - 1747589185*x^16 - 2338480852*x^15 + 719039461*x^14 + 702585559*x^13 - 204971923*x^12 - 140848481*x^11 + 39126498*x^10 + 18138585*x^9 - 4764404*x^8 - 1426972*x^7 + 345013*x^6 + 64118*x^5 - 13350*x^4 - 1490*x^3 + 225*x^2 + 15*x - 1);
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))];