\\ Pari/GP code for working with number field 20.4.281124869512857892173749682176.11.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^20 - 10*y^19 + 62*y^18 - 256*y^17 + 783*y^16 - 1774*y^15 + 2925*y^14 - 3038*y^13 + 372*y^12 + 5766*y^11 - 13472*y^10 + 18618*y^9 - 17947*y^8 + 12464*y^7 - 6281*y^6 + 2078*y^5 + 48*y^4 - 684*y^3 + 487*y^2 - 164*y + 23, 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^20 - 10*x^19 + 62*x^18 - 256*x^17 + 783*x^16 - 1774*x^15 + 2925*x^14 - 3038*x^13 + 372*x^12 + 5766*x^11 - 13472*x^10 + 18618*x^9 - 17947*x^8 + 12464*x^7 - 6281*x^6 + 2078*x^5 + 48*x^4 - 684*x^3 + 487*x^2 - 164*x + 23, 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]])