\\ Pari/GP code for working with number field 20.20.4884274415246183319344904689389832005326536704.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^20 - 10*y^19 - 116*y^18 + 1313*y^17 + 4736*y^16 - 67464*y^15 - 71774*y^14 + 1744748*y^13 - 126850*y^12 - 24570618*y^11 + 14939336*y^10 + 193331394*y^9 - 163802315*y^8 - 826609978*y^7 + 787082462*y^6 + 1707884929*y^5 - 1813914636*y^4 - 1179787402*y^3 + 1644305872*y^2 - 315124556*y - 58612856, 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 - 10*x^19 - 116*x^18 + 1313*x^17 + 4736*x^16 - 67464*x^15 - 71774*x^14 + 1744748*x^13 - 126850*x^12 - 24570618*x^11 + 14939336*x^10 + 193331394*x^9 - 163802315*x^8 - 826609978*x^7 + 787082462*x^6 + 1707884929*x^5 - 1813914636*x^4 - 1179787402*x^3 + 1644305872*x^2 - 315124556*x - 58612856, 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]])