\\ Pari/GP code for working with number field 20.12.88312074181571967846984863281250000.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^20 - 6*y^19 - y^18 + 62*y^17 - 72*y^16 - 619*y^15 + 1808*y^14 + 2586*y^13 - 17551*y^12 + 18084*y^11 + 51190*y^10 - 138959*y^9 + 17681*y^8 + 250533*y^7 - 232783*y^6 - 82051*y^5 + 268062*y^4 - 129551*y^3 - 69134*y^2 + 78266*y - 18031, 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^20 - 6*x^19 - x^18 + 62*x^17 - 72*x^16 - 619*x^15 + 1808*x^14 + 2586*x^13 - 17551*x^12 + 18084*x^11 + 51190*x^10 - 138959*x^9 + 17681*x^8 + 250533*x^7 - 232783*x^6 - 82051*x^5 + 268062*x^4 - 129551*x^3 - 69134*x^2 + 78266*x - 18031, 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]])