\\ Pari/GP code for working with number field 20.4.288585859854484970400390625.1.
\\ 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 + 14*y^18 - 21*y^17 + 40*y^16 - 60*y^15 + 3*y^14 + 126*y^13 - 230*y^12 + 382*y^11 - 570*y^10 + 197*y^9 + 981*y^8 - 1528*y^7 + 265*y^6 + 1238*y^5 - 1018*y^4 - 168*y^3 + 564*y^2 - 220*y + 11, 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 - 6*x^19 + 14*x^18 - 21*x^17 + 40*x^16 - 60*x^15 + 3*x^14 + 126*x^13 - 230*x^12 + 382*x^11 - 570*x^10 + 197*x^9 + 981*x^8 - 1528*x^7 + 265*x^6 + 1238*x^5 - 1018*x^4 - 168*x^3 + 564*x^2 - 220*x + 11, 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]])