\\ Pari/GP code for working with number field 22.14.1275054393023762909963506622668395857607199180560057153446822299847849103680658423408180099481492815135308821561344.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^22 - 242454*y^20 - 66254755944*y^18 + 9988149125214624*y^16 + 2412251307077297217168*y^14 - 97126283753436220637656848*y^12 - 36043439335840553783024429840880*y^10 - 968324930877205693808500678923025440*y^8 + 411693823788000055227773052463075223008512*y^6 - 14674638475171337576102371060253129696887988224*y^4 + 164949721485379801300637463025582558350584505753600*y^2 - 571411117069770362003780702945732790998630818613760000, 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^22 - 242454*x^20 - 66254755944*x^18 + 9988149125214624*x^16 + 2412251307077297217168*x^14 - 97126283753436220637656848*x^12 - 36043439335840553783024429840880*x^10 - 968324930877205693808500678923025440*x^8 + 411693823788000055227773052463075223008512*x^6 - 14674638475171337576102371060253129696887988224*x^4 + 164949721485379801300637463025582558350584505753600*x^2 - 571411117069770362003780702945732790998630818613760000, 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]])