\\ Pari/GP code for working with number field 21.1.569752580450182296234409142462680787211686353567744.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^21 - 154*y^19 - 350*y^18 + 9422*y^17 + 38318*y^16 - 232820*y^15 - 1495990*y^14 - 59192*y^13 + 16100140*y^12 + 74605944*y^11 + 291794048*y^10 + 22521464*y^9 - 4000275328*y^8 - 1691077124*y^7 + 11054922536*y^6 - 418374125568*y^5 - 2346556141272*y^4 - 4154662343608*y^3 - 3218488878808*y^2 - 17560545695056*y - 50542023253032, 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

\\ 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^21 - 154*x^19 - 350*x^18 + 9422*x^17 + 38318*x^16 - 232820*x^15 - 1495990*x^14 - 59192*x^13 + 16100140*x^12 + 74605944*x^11 + 291794048*x^10 + 22521464*x^9 - 4000275328*x^8 - 1691077124*x^7 + 11054922536*x^6 - 418374125568*x^5 - 2346556141272*x^4 - 4154662343608*x^3 - 3218488878808*x^2 - 17560545695056*x - 50542023253032, 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]])