// Magma code for working with number field 32.0.2088443876129429457733048543333054873029337200425307489036314630977400864768.1

// 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^32 + 416*x^30 + 78416*x^28 + 8858304*x^26 + 668327400*x^24 + 35525314240*x^22 + 1367724598240*x^20 + 38608911516032*x^18 + 799928385472788*x^16 + 12056891607126080*x^14 + 129666388829365024*x^12 + 963236031303854464*x^10 + 4695775652606290512*x^8 + 13840180870839593088*x^6 + 21419327538204132160*x^4 + 13103588611607233792*x^2 + 1330833218366359682);

// 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^32 + 416*x^30 + 78416*x^28 + 8858304*x^26 + 668327400*x^24 + 35525314240*x^22 + 1367724598240*x^20 + 38608911516032*x^18 + 799928385472788*x^16 + 12056891607126080*x^14 + 129666388829365024*x^12 + 963236031303854464*x^10 + 4695775652606290512*x^8 + 13840180870839593088*x^6 + 21419327538204132160*x^4 + 13103588611607233792*x^2 + 1330833218366359682);
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))];