\\ Pari/GP code for working with number field 22.2.1767712543554828434373148672.1

\\ Some of these functions may take a long time to execute (this depends on the field).

\\ Define the number field: 
K = bnfinit(y^22 - 5*y^21 + 17*y^20 - 40*y^19 + 70*y^18 - 93*y^17 + 99*y^16 - 86*y^15 + 53*y^14 + 37*y^13 - 283*y^12 + 778*y^11 - 1493*y^10 + 2198*y^9 - 2575*y^8 + 2425*y^7 - 1840*y^6 + 1108*y^5 - 512*y^4 + 173*y^3 - 33*y^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 - 5*x^21 + 17*x^20 - 40*x^19 + 70*x^18 - 93*x^17 + 99*x^16 - 86*x^15 + 53*x^14 + 37*x^13 - 283*x^12 + 778*x^11 - 1493*x^10 + 2198*x^9 - 2575*x^8 + 2425*x^7 - 1840*x^6 + 1108*x^5 - 512*x^4 + 173*x^3 - 33*x^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(b)]

\\ Galois group: 
polgalois(K.pol)

\\ Frobenius cycle types: 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])