// Magma code for working with number field 45.45.690502119755999041650933728181031670984689765738417760130764409811234288886886833270367975576584121365281.1

// 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 - 146*x^43 + 698*x^42 + 8985*x^41 - 51888*x^40 - 296427*x^39 + 2177976*x^38 + 5305247*x^37 - 57476954*x^36 - 34012820*x^35 + 1002110429*x^34 - 626632964*x^33 - 11753633659*x^32 + 18688879395*x^31 + 91599636078*x^30 - 237738671647*x^29 - 440115277508*x^28 + 1861290436646*x^27 + 876346054868*x^26 - 9636797401413*x^25 + 3638557275086*x^24 + 33031864898329*x^23 - 35155604700368*x^22 - 70196515943930*x^21 + 133850962002890*x^20 + 68063148815178*x^19 - 296060971462243*x^18 + 63115875057640*x^17 + 386991855136149*x^16 - 292070548579474*x^15 - 252378957195025*x^14 + 393110942284862*x^13 - 3989425272418*x^12 - 248373295675609*x^11 + 116468486134606*x^10 + 55635812822046*x^9 - 63558461838326*x^8 + 10039317831256*x^7 + 9501878099079*x^6 - 4834005607020*x^5 + 527330092663*x^4 + 166660835109*x^3 - 51025747438*x^2 + 4407259697*x - 98524081);

// 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 - 146*x^43 + 698*x^42 + 8985*x^41 - 51888*x^40 - 296427*x^39 + 2177976*x^38 + 5305247*x^37 - 57476954*x^36 - 34012820*x^35 + 1002110429*x^34 - 626632964*x^33 - 11753633659*x^32 + 18688879395*x^31 + 91599636078*x^30 - 237738671647*x^29 - 440115277508*x^28 + 1861290436646*x^27 + 876346054868*x^26 - 9636797401413*x^25 + 3638557275086*x^24 + 33031864898329*x^23 - 35155604700368*x^22 - 70196515943930*x^21 + 133850962002890*x^20 + 68063148815178*x^19 - 296060971462243*x^18 + 63115875057640*x^17 + 386991855136149*x^16 - 292070548579474*x^15 - 252378957195025*x^14 + 393110942284862*x^13 - 3989425272418*x^12 - 248373295675609*x^11 + 116468486134606*x^10 + 55635812822046*x^9 - 63558461838326*x^8 + 10039317831256*x^7 + 9501878099079*x^6 - 4834005607020*x^5 + 527330092663*x^4 + 166660835109*x^3 - 51025747438*x^2 + 4407259697*x - 98524081);
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))];