\\ Pari/GP code for working with number field 20.20.516853980088694624404163683037236690944.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^20 - 83*y^18 - 108*y^17 + 2380*y^16 + 5760*y^15 - 27524*y^14 - 103968*y^13 + 77348*y^12 + 735960*y^11 + 616108*y^10 - 1615050*y^9 - 3200232*y^8 - 714960*y^7 + 2411685*y^6 + 1666554*y^5 - 423490*y^4 - 635688*y^3 - 65073*y^2 + 67764*y + 15313, 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 - 83*x^18 - 108*x^17 + 2380*x^16 + 5760*x^15 - 27524*x^14 - 103968*x^13 + 77348*x^12 + 735960*x^11 + 616108*x^10 - 1615050*x^9 - 3200232*x^8 - 714960*x^7 + 2411685*x^6 + 1666554*x^5 - 423490*x^4 - 635688*x^3 - 65073*x^2 + 67764*x + 15313, 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]])