// Magma code for working with number field 45.15.1744174787462733669236743003323349838789896331385003649618101010724691.1

// 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 - 12*x^44 + 42*x^43 + 36*x^42 - 525*x^41 + 702*x^40 + 2050*x^39 - 5802*x^38 - 2832*x^37 + 26919*x^36 - 20811*x^35 - 75546*x^34 + 201699*x^33 - 97125*x^32 - 532302*x^31 + 1289591*x^30 - 477339*x^29 - 2405610*x^28 + 4128988*x^27 - 1666824*x^26 - 5259345*x^25 + 9273382*x^24 + 3749217*x^23 - 15223818*x^22 - 9661128*x^21 + 33692574*x^20 - 4282803*x^19 - 38096273*x^18 + 36480543*x^17 - 14154921*x^16 + 10925288*x^15 - 437979*x^14 - 36804246*x^13 + 57064132*x^12 - 25852146*x^11 - 34300866*x^10 + 48986404*x^9 - 2084457*x^8 - 23365995*x^7 + 5433983*x^6 + 5348430*x^5 - 1230861*x^4 - 665734*x^3 + 68376*x^2 + 38973*x + 2969);

// 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 - 12*x^44 + 42*x^43 + 36*x^42 - 525*x^41 + 702*x^40 + 2050*x^39 - 5802*x^38 - 2832*x^37 + 26919*x^36 - 20811*x^35 - 75546*x^34 + 201699*x^33 - 97125*x^32 - 532302*x^31 + 1289591*x^30 - 477339*x^29 - 2405610*x^28 + 4128988*x^27 - 1666824*x^26 - 5259345*x^25 + 9273382*x^24 + 3749217*x^23 - 15223818*x^22 - 9661128*x^21 + 33692574*x^20 - 4282803*x^19 - 38096273*x^18 + 36480543*x^17 - 14154921*x^16 + 10925288*x^15 - 437979*x^14 - 36804246*x^13 + 57064132*x^12 - 25852146*x^11 - 34300866*x^10 + 48986404*x^9 - 2084457*x^8 - 23365995*x^7 + 5433983*x^6 + 5348430*x^5 - 1230861*x^4 - 665734*x^3 + 68376*x^2 + 38973*x + 2969);
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))];