// Magma code for working with number field 24.4.61343664831565854732948944775316051495075225830078125.2.
// 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^24 - 35*x^22 - 55*x^21 + 955*x^20 + 505*x^19 - 13360*x^18 - 20130*x^17 + 209535*x^16 + 359985*x^15 - 3499870*x^14 + 3872145*x^13 + 4149075*x^12 + 14576245*x^11 - 73943310*x^10 - 41050830*x^9 + 549395715*x^8 - 939702525*x^7 + 1082823105*x^6 - 269406495*x^5 - 9880462205*x^4 + 31459457800*x^3 - 44701823080*x^2 + 26766458640*x - 4828553680);

// 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(Rationals()); K<a> := NumberField(x^24 - 35*x^22 - 55*x^21 + 955*x^20 + 505*x^19 - 13360*x^18 - 20130*x^17 + 209535*x^16 + 359985*x^15 - 3499870*x^14 + 3872145*x^13 + 4149075*x^12 + 14576245*x^11 - 73943310*x^10 - 41050830*x^9 + 549395715*x^8 - 939702525*x^7 + 1082823105*x^6 - 269406495*x^5 - 9880462205*x^4 + 31459457800*x^3 - 44701823080*x^2 + 26766458640*x - 4828553680);
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 pO_K for p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];