\\ Pari/GP code for working with number field 22.4.120805734981681461698275794063.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^22 - 3*y^21 + 2*y^20 + 6*y^19 - 15*y^18 + 11*y^17 + 10*y^16 - 31*y^15 + 28*y^14 + 2*y^13 - 32*y^12 + 35*y^11 - 11*y^10 - 16*y^9 + 25*y^8 - 14*y^7 - 4*y^6 + 10*y^5 - 8*y^4 + 2*y^3 + 2*y^2 - 2*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^22 - 3*x^21 + 2*x^20 + 6*x^19 - 15*x^18 + 11*x^17 + 10*x^16 - 31*x^15 + 28*x^14 + 2*x^13 - 32*x^12 + 35*x^11 - 11*x^10 - 16*x^9 + 25*x^8 - 14*x^7 - 4*x^6 + 10*x^5 - 8*x^4 + 2*x^3 + 2*x^2 - 2*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(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]])