\\ Pari/GP code for working with number field 16.8.2164959798672044689794137876838502993.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^16 - 4*y^15 - 41*y^14 - 130*y^13 - 2186*y^12 + 10546*y^11 + 124718*y^10 + 77955*y^9 + 18509*y^8 + 8606523*y^7 + 27735607*y^6 + 128700291*y^5 + 1368635976*y^4 + 4658216687*y^3 + 3073249780*y^2 - 7839808132*y - 9179159021, 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 - 4*x^15 - 41*x^14 - 130*x^13 - 2186*x^12 + 10546*x^11 + 124718*x^10 + 77955*x^9 + 18509*x^8 + 8606523*x^7 + 27735607*x^6 + 128700291*x^5 + 1368635976*x^4 + 4658216687*x^3 + 3073249780*x^2 - 7839808132*x - 9179159021, 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]])