\\ Pari/GP code for working with number field 20.0.934801626748320922851562500000000000000000000.7.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^20 + 5*y^18 - 20*y^17 + 910*y^16 + 236*y^15 + 26360*y^14 + 26460*y^13 + 737860*y^12 + 804900*y^11 + 15113210*y^10 + 12037680*y^9 + 241238600*y^8 + 119368900*y^7 + 2830853065*y^6 + 775988298*y^5 + 23347784530*y^4 + 3610501410*y^3 + 122436468035*y^2 + 11447764160*y + 315569310049, 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 + 5*x^18 - 20*x^17 + 910*x^16 + 236*x^15 + 26360*x^14 + 26460*x^13 + 737860*x^12 + 804900*x^11 + 15113210*x^10 + 12037680*x^9 + 241238600*x^8 + 119368900*x^7 + 2830853065*x^6 + 775988298*x^5 + 23347784530*x^4 + 3610501410*x^3 + 122436468035*x^2 + 11447764160*x + 315569310049, 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]])