// Magma code for working with number field 45.45.40844008098536976898528926596491646392449447921859499771891413268750045487853347665634360111802937199321.1

// 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 - 3*x^44 - 126*x^43 + 350*x^42 + 7095*x^41 - 18207*x^40 - 237254*x^39 + 560949*x^38 + 5280957*x^37 - 11465163*x^36 - 83127609*x^35 + 164960568*x^34 + 959694733*x^33 - 1729668720*x^32 - 8316855543*x^31 + 13495839126*x^30 + 54911329452*x^29 - 79318322604*x^28 - 278624618803*x^27 + 353264598507*x^26 + 1090386630918*x^25 - 1193881051406*x^24 - 3286918329246*x^23 + 3054422504946*x^22 + 7584376524784*x^21 - 5885779838802*x^20 - 13239515051214*x^19 + 8489225140885*x^18 + 17170281215082*x^17 - 9118144480932*x^16 - 16115327765898*x^15 + 7281918623109*x^14 + 10535410010724*x^13 - 4323241543716*x^12 - 4518883017183*x^11 + 1877328461526*x^10 + 1138556965481*x^9 - 554295751305*x^8 - 124509763362*x^7 + 91825606123*x^6 - 3656082528*x^5 - 5240896776*x^4 + 961503906*x^3 - 1541688*x^2 - 11847522*x + 756289);

// 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 - 3*x^44 - 126*x^43 + 350*x^42 + 7095*x^41 - 18207*x^40 - 237254*x^39 + 560949*x^38 + 5280957*x^37 - 11465163*x^36 - 83127609*x^35 + 164960568*x^34 + 959694733*x^33 - 1729668720*x^32 - 8316855543*x^31 + 13495839126*x^30 + 54911329452*x^29 - 79318322604*x^28 - 278624618803*x^27 + 353264598507*x^26 + 1090386630918*x^25 - 1193881051406*x^24 - 3286918329246*x^23 + 3054422504946*x^22 + 7584376524784*x^21 - 5885779838802*x^20 - 13239515051214*x^19 + 8489225140885*x^18 + 17170281215082*x^17 - 9118144480932*x^16 - 16115327765898*x^15 + 7281918623109*x^14 + 10535410010724*x^13 - 4323241543716*x^12 - 4518883017183*x^11 + 1877328461526*x^10 + 1138556965481*x^9 - 554295751305*x^8 - 124509763362*x^7 + 91825606123*x^6 - 3656082528*x^5 - 5240896776*x^4 + 961503906*x^3 - 1541688*x^2 - 11847522*x + 756289);
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))];