\\ Pari/GP code for working with number field 20.10.815119333245299093090410107421875.2.
\\ 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 + 3*y^18 + 45*y^17 - 93*y^16 + 148*y^15 - 539*y^14 - 489*y^13 + 3422*y^12 - 894*y^11 + 2973*y^10 - 11008*y^9 + 12614*y^8 - 54399*y^7 + 47350*y^6 + 33765*y^5 - 16959*y^4 - 2255*y^3 - 5890*y^2 + 1605*y + 605, 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 + 3*x^18 + 45*x^17 - 93*x^16 + 148*x^15 - 539*x^14 - 489*x^13 + 3422*x^12 - 894*x^11 + 2973*x^10 - 11008*x^9 + 12614*x^8 - 54399*x^7 + 47350*x^6 + 33765*x^5 - 16959*x^4 - 2255*x^3 - 5890*x^2 + 1605*x + 605, 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]])