\\ Pari/GP code for working with number field 16.4.557121553975260619997184.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^16 + 16*y^14 - 12*y^13 + 78*y^12 - 40*y^11 - 44*y^10 + 568*y^9 - 2088*y^8 + 4112*y^7 - 5620*y^6 + 6192*y^5 - 7528*y^4 + 9152*y^3 - 7464*y^2 + 3264*y - 578, 1)

\\ Defining polynomial: 
K.pol

\\ Degree over Q: 
poldegree(K.pol)

\\ Signature: 
K.sign

\\ Discriminant: 
K.disc

\\ Ramified primes: 
factor(abs(K.disc))[,1]~

\\ Integral basis: 
K.zk

\\ Class group: 
K.clgp

\\ Unit rank: 
K.fu

\\ Generator for roots of unity: 
K.tu[2]

\\ Fundamental units: 
K.fu

\\ Regulator: 
K.reg

\\ Analytic class number formula: 
\\ self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^16 + 16*x^14 - 12*x^13 + 78*x^12 - 40*x^11 - 44*x^10 + 568*x^9 - 2088*x^8 + 4112*x^7 - 5620*x^6 + 6192*x^5 - 7528*x^4 + 9152*x^3 - 7464*x^2 + 3264*x - 578, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

\\ Intermediate fields: 
L = nfsubfields(K); L[2..length(L)]

\\ Galois group: 
polgalois(K.pol)

\\ Frobenius cycle types: 
\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])