// Magma code for working with number field 32.32.2283054469939826689646603434893544205211768622361793362696380065480141214463295488.1

// 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 - 992*x^30 + 445904*x^28 - 120117312*x^26 + 21610391400*x^24 - 2739237167680*x^22 + 251483043048160*x^20 - 16928401412041856*x^18 + 836368832263692948*x^16 - 30060792811796500160*x^14 + 770922695655253881376*x^12 - 13656344894464497327232*x^10 + 158755009398149781429072*x^8 - 1115784195608048666238336*x^6 + 4117775007601131982546240*x^4 - 6007107069912239598067456*x^2 + 1454846243494370527656962);

// 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 - 992*x^30 + 445904*x^28 - 120117312*x^26 + 21610391400*x^24 - 2739237167680*x^22 + 251483043048160*x^20 - 16928401412041856*x^18 + 836368832263692948*x^16 - 30060792811796500160*x^14 + 770922695655253881376*x^12 - 13656344894464497327232*x^10 + 158755009398149781429072*x^8 - 1115784195608048666238336*x^6 + 4117775007601131982546240*x^4 - 6007107069912239598067456*x^2 + 1454846243494370527656962);
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))];