\\ Pari/GP code for working with number field 21.21.34061631211994616985595974961974214656.2.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^21 - 42*y^19 + 756*y^17 - 7616*y^15 - 32*y^14 + 47040*y^13 + 896*y^12 - 183456*y^11 - 9856*y^10 + 448448*y^9 + 53760*y^8 - 658368*y^7 - 150528*y^6 + 524160*y^5 + 200704*y^4 - 164864*y^3 - 100352*y^2 - 10752*y + 512, 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 - 42*x^19 + 756*x^17 - 7616*x^15 - 32*x^14 + 47040*x^13 + 896*x^12 - 183456*x^11 - 9856*x^10 + 448448*x^9 + 53760*x^8 - 658368*x^7 - 150528*x^6 + 524160*x^5 + 200704*x^4 - 164864*x^3 - 100352*x^2 - 10752*x + 512, 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]])