\\ Pari/GP code for working with number field 16.8.9271430660151733401059603251196113.1

\\ Some of these functions may take a long time to execute (this depends on the field).

\\ Define the number field: 
K = bnfinit(y^16 - y^15 - 45*y^14 + y^13 - 708*y^12 + 1115*y^11 + 16242*y^10 + 36896*y^9 + 185901*y^8 - 544814*y^7 - 1337386*y^6 - 7688656*y^5 - 9683921*y^4 + 17611964*y^3 + 39000939*y^2 + 15688356*y - 1027001, 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 - x^15 - 45*x^14 + x^13 - 708*x^12 + 1115*x^11 + 16242*x^10 + 36896*x^9 + 185901*x^8 - 544814*x^7 - 1337386*x^6 - 7688656*x^5 - 9683921*x^4 + 17611964*x^3 + 39000939*x^2 + 15688356*x - 1027001, 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(b)]

\\ Galois group: 
polgalois(K.pol)

\\ 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 Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])