\\ Pari/GP code for working with number field 20.0.4141864704128763454804479176602577540516903124992.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^20 - 4*y^19 + 177*y^18 - 4*y^17 + 8449*y^16 + 23822*y^15 + 168540*y^14 + 596526*y^13 + 2612317*y^12 + 7067238*y^11 + 38130743*y^10 + 75736410*y^9 + 321304814*y^8 + 864149074*y^7 + 1787066994*y^6 + 245640690*y^5 + 6843858589*y^4 - 4594563420*y^3 + 14977761454*y^2 - 24323038390*y + 15554125329, 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 - 4*x^19 + 177*x^18 - 4*x^17 + 8449*x^16 + 23822*x^15 + 168540*x^14 + 596526*x^13 + 2612317*x^12 + 7067238*x^11 + 38130743*x^10 + 75736410*x^9 + 321304814*x^8 + 864149074*x^7 + 1787066994*x^6 + 245640690*x^5 + 6843858589*x^4 - 4594563420*x^3 + 14977761454*x^2 - 24323038390*x + 15554125329, 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]])