// Magma code for working with number field 45.45.16527441490674381067348948889146522048718287580785548225593089356497792068001085725808918264262962961.1

// 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 - 114*x^43 + 468*x^42 + 5696*x^41 - 24132*x^40 - 165566*x^39 + 729869*x^38 + 3130045*x^37 - 14526130*x^36 - 40702810*x^35 + 202181480*x^34 + 374739344*x^33 - 2040701711*x^32 - 2468141099*x^31 + 15280860430*x^30 + 11523689449*x^29 - 86102308596*x^28 - 36476021474*x^27 + 368042106614*x^26 + 66266177672*x^25 - 1197228116258*x^24 + 511532340*x^23 + 2959992694385*x^22 - 394556499523*x^21 - 5529580093472*x^20 + 1265687493078*x^19 + 7719490702430*x^18 - 2271295840968*x^17 - 7914078560767*x^16 + 2645087732416*x^15 + 5804631677810*x^14 - 2048306537737*x^13 - 2929193071000*x^12 + 1039444940408*x^11 + 957147248584*x^10 - 333732759991*x^9 - 182597145321*x^8 + 64595300484*x^7 + 16257686900*x^6 - 6916885901*x^5 - 166352258*x^4 + 298954277*x^3 - 40977465*x^2 + 2118031*x - 36851);

// 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 - 114*x^43 + 468*x^42 + 5696*x^41 - 24132*x^40 - 165566*x^39 + 729869*x^38 + 3130045*x^37 - 14526130*x^36 - 40702810*x^35 + 202181480*x^34 + 374739344*x^33 - 2040701711*x^32 - 2468141099*x^31 + 15280860430*x^30 + 11523689449*x^29 - 86102308596*x^28 - 36476021474*x^27 + 368042106614*x^26 + 66266177672*x^25 - 1197228116258*x^24 + 511532340*x^23 + 2959992694385*x^22 - 394556499523*x^21 - 5529580093472*x^20 + 1265687493078*x^19 + 7719490702430*x^18 - 2271295840968*x^17 - 7914078560767*x^16 + 2645087732416*x^15 + 5804631677810*x^14 - 2048306537737*x^13 - 2929193071000*x^12 + 1039444940408*x^11 + 957147248584*x^10 - 333732759991*x^9 - 182597145321*x^8 + 64595300484*x^7 + 16257686900*x^6 - 6916885901*x^5 - 166352258*x^4 + 298954277*x^3 - 40977465*x^2 + 2118031*x - 36851);
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))];