\\ Pari/GP code for working with number field 32.0.343518132400968646195554032900431766403471261479800406016.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^32 - 30*y^30 + 536*y^28 - 6344*y^26 + 55936*y^24 - 372416*y^22 + 1935424*y^20 - 7766016*y^18 + 24284672*y^16 - 57272832*y^14 + 101765120*y^12 - 126291968*y^10 + 110481408*y^8 - 54362112*y^6 + 17793024*y^4 - 1179648*y^2 + 65536, 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^32 - 30*x^30 + 536*x^28 - 6344*x^26 + 55936*x^24 - 372416*x^22 + 1935424*x^20 - 7766016*x^18 + 24284672*x^16 - 57272832*x^14 + 101765120*x^12 - 126291968*x^10 + 110481408*x^8 - 54362112*x^6 + 17793024*x^4 - 1179648*x^2 + 65536, 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]])