\\ Pari/GP code for working with number field 20.16.55205220243101566254904697284680320000.2.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^20 - 2*y^19 - 58*y^18 - 29*y^17 + 601*y^16 - 348*y^15 - 3815*y^14 + 4392*y^13 + 9768*y^12 - 7124*y^11 + 4105*y^10 - 60836*y^9 - 11960*y^8 + 223601*y^7 - 101287*y^6 - 209801*y^5 + 152847*y^4 + 35810*y^3 - 34168*y^2 - 2199*y + 503, 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 - 2*x^19 - 58*x^18 - 29*x^17 + 601*x^16 - 348*x^15 - 3815*x^14 + 4392*x^13 + 9768*x^12 - 7124*x^11 + 4105*x^10 - 60836*x^9 - 11960*x^8 + 223601*x^7 - 101287*x^6 - 209801*x^5 + 152847*x^4 + 35810*x^3 - 34168*x^2 - 2199*x + 503, 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]])