// Magma code for working with number field 45.45.1136296607981427246297191641801603901004077832833182160525696498824571613560438852133526835484889.1

// 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 - 38*x^43 + 936*x^42 - 488*x^41 - 32994*x^40 + 61155*x^39 + 695151*x^38 - 1826452*x^37 - 9745559*x^36 + 31398650*x^35 + 95576968*x^34 - 363031278*x^33 - 668271620*x^32 + 3006554151*x^31 + 3299628259*x^30 - 18405469866*x^29 - 10819187452*x^28 + 84606290574*x^27 + 17798672756*x^26 - 293624042239*x^25 + 25779977473*x^24 + 766879229695*x^23 - 259704306954*x^22 - 1488735953755*x^21 + 843265985246*x^20 + 2094958274346*x^19 - 1672596038928*x^18 - 2036992655261*x^17 + 2211683013511*x^16 + 1229593352332*x^15 - 1960884055901*x^14 - 305227614428*x^13 + 1128154934100*x^12 - 126915280282*x^11 - 391023252143*x^10 + 124192084723*x^9 + 69369604379*x^8 - 37452923715*x^7 - 3334757683*x^6 + 4610823573*x^5 - 413774775*x^4 - 188088413*x^3 + 35065204*x^2 - 872335*x + 619);

// 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 - 38*x^43 + 936*x^42 - 488*x^41 - 32994*x^40 + 61155*x^39 + 695151*x^38 - 1826452*x^37 - 9745559*x^36 + 31398650*x^35 + 95576968*x^34 - 363031278*x^33 - 668271620*x^32 + 3006554151*x^31 + 3299628259*x^30 - 18405469866*x^29 - 10819187452*x^28 + 84606290574*x^27 + 17798672756*x^26 - 293624042239*x^25 + 25779977473*x^24 + 766879229695*x^23 - 259704306954*x^22 - 1488735953755*x^21 + 843265985246*x^20 + 2094958274346*x^19 - 1672596038928*x^18 - 2036992655261*x^17 + 2211683013511*x^16 + 1229593352332*x^15 - 1960884055901*x^14 - 305227614428*x^13 + 1128154934100*x^12 - 126915280282*x^11 - 391023252143*x^10 + 124192084723*x^9 + 69369604379*x^8 - 37452923715*x^7 - 3334757683*x^6 + 4610823573*x^5 - 413774775*x^4 - 188088413*x^3 + 35065204*x^2 - 872335*x + 619);
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))];