\\ Pari/GP code for working with number field 16.0.2371212900396504113756570569.4.
\\ 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 + 26*y^14 - 47*y^13 + 225*y^12 - 421*y^11 + 1143*y^10 - 585*y^9 + 2609*y^8 - 2538*y^7 + 8158*y^6 - 3788*y^5 + 2318*y^4 + 1389*y^3 + 1575*y^2 + 459*y + 81, 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

\\ Narrow class group: 
bnfnarrow(K)

\\ 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 + 26*x^14 - 47*x^13 + 225*x^12 - 421*x^11 + 1143*x^10 - 585*x^9 + 2609*x^8 - 2538*x^7 + 8158*x^6 - 3788*x^5 + 2318*x^4 + 1389*x^3 + 1575*x^2 + 459*x + 81, 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]])