\\ Pari/GP code for working with number field 22.0.398976882084318595545873356311326226309728822028879.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^22 - y^21 + 203*y^20 + 241*y^19 + 15645*y^18 + 49907*y^17 + 645544*y^16 + 2979464*y^15 + 17383054*y^14 + 82563250*y^13 + 322791094*y^12 + 1212821214*y^11 + 3698021698*y^10 + 9684850830*y^9 + 21693864639*y^8 + 38502084388*y^7 + 50771931117*y^6 + 47768698105*y^5 + 38829007897*y^4 + 30616690773*y^3 + 52753103834*y^2 + 67713149325*y + 81132029807, 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^22 - x^21 + 203*x^20 + 241*x^19 + 15645*x^18 + 49907*x^17 + 645544*x^16 + 2979464*x^15 + 17383054*x^14 + 82563250*x^13 + 322791094*x^12 + 1212821214*x^11 + 3698021698*x^10 + 9684850830*x^9 + 21693864639*x^8 + 38502084388*x^7 + 50771931117*x^6 + 47768698105*x^5 + 38829007897*x^4 + 30616690773*x^3 + 52753103834*x^2 + 67713149325*x + 81132029807, 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]])