\\ Pari/GP code for working with number field 20.4.688077672624913964532736.1.
\\ 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 + 19*y^18 - 51*y^17 + 109*y^16 - 167*y^15 + 218*y^14 - 255*y^13 + 201*y^12 - 84*y^11 - 43*y^10 + 137*y^9 - 68*y^8 + 32*y^7 - 81*y^6 + 17*y^5 - y^4 + 15*y^3 + y^2 + 6*y + 1, 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 + 19*x^18 - 51*x^17 + 109*x^16 - 167*x^15 + 218*x^14 - 255*x^13 + 201*x^12 - 84*x^11 - 43*x^10 + 137*x^9 - 68*x^8 + 32*x^7 - 81*x^6 + 17*x^5 - x^4 + 15*x^3 + x^2 + 6*x + 1, 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]])