\\ Pari/GP code for working with number field 21.13.40453417591156198845459427294648145741.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^21 - 21*y^19 - 3*y^18 + 186*y^17 + 45*y^16 - 907*y^15 - 270*y^14 + 2685*y^13 + 854*y^12 - 5055*y^11 - 1653*y^10 + 6124*y^9 + 2244*y^8 - 4458*y^7 - 2040*y^6 + 1560*y^5 + 960*y^4 - 80*y^3 - 123*y^2 - 21*y - 1, 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 - 21*x^19 - 3*x^18 + 186*x^17 + 45*x^16 - 907*x^15 - 270*x^14 + 2685*x^13 + 854*x^12 - 5055*x^11 - 1653*x^10 + 6124*x^9 + 2244*x^8 - 4458*x^7 - 2040*x^6 + 1560*x^5 + 960*x^4 - 80*x^3 - 123*x^2 - 21*x - 1, 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(b)]

\\ 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]])