// Magma code for working with number field 45.45.940750991812442660132286068374700929301371405411689329186316523736000865610069804102171689.1

// 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 - 81*x^43 + 2943*x^41 - 63576*x^39 + 912699*x^37 - 256*x^36 - 9221688*x^35 + 11619*x^34 + 67767147*x^33 - 228393*x^32 - 369266337*x^31 + 2577030*x^30 + 1508018391*x^29 - 18638901*x^28 - 4636251652*x^27 + 91308222*x^26 + 10725370218*x^25 - 311639850*x^24 - 18578308230*x^23 + 749536389*x^22 + 23873708220*x^21 - 1268865972*x^20 - 22446274041*x^19 + 1494154410*x^18 + 15154469253*x^17 - 1197712170*x^16 - 7173383040*x^15 + 633986568*x^14 + 2312491545*x^13 - 213869688*x^12 - 490418469*x^11 + 44511588*x^10 + 65361426*x^9 - 5507676*x^8 - 5123475*x^7 + 385695*x^6 + 214380*x^5 - 14391*x^4 - 4176*x^3 + 243*x^2 + 27*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 - 81*x^43 + 2943*x^41 - 63576*x^39 + 912699*x^37 - 256*x^36 - 9221688*x^35 + 11619*x^34 + 67767147*x^33 - 228393*x^32 - 369266337*x^31 + 2577030*x^30 + 1508018391*x^29 - 18638901*x^28 - 4636251652*x^27 + 91308222*x^26 + 10725370218*x^25 - 311639850*x^24 - 18578308230*x^23 + 749536389*x^22 + 23873708220*x^21 - 1268865972*x^20 - 22446274041*x^19 + 1494154410*x^18 + 15154469253*x^17 - 1197712170*x^16 - 7173383040*x^15 + 633986568*x^14 + 2312491545*x^13 - 213869688*x^12 - 490418469*x^11 + 44511588*x^10 + 65361426*x^9 - 5507676*x^8 - 5123475*x^7 + 385695*x^6 + 214380*x^5 - 14391*x^4 - 4176*x^3 + 243*x^2 + 27*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))];