\\ Pari/GP code for working with number field 23.23.542693874230042671882983450092579839839306394717839529.1

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

\\ Define the number field: 
K = bnfinit(y^23 - y^22 - 132*y^21 + 269*y^20 + 6722*y^19 - 19118*y^18 - 167488*y^17 + 601226*y^16 + 2127300*y^15 - 9664345*y^14 - 12698450*y^13 + 83446410*y^12 + 19888027*y^11 - 390244377*y^10 + 110929757*y^9 + 994369765*y^8 - 511651057*y^7 - 1370318108*y^6 + 775297206*y^5 + 972267257*y^4 - 466392148*y^3 - 312747839*y^2 + 85444683*y + 38737459, 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^23 - x^22 - 132*x^21 + 269*x^20 + 6722*x^19 - 19118*x^18 - 167488*x^17 + 601226*x^16 + 2127300*x^15 - 9664345*x^14 - 12698450*x^13 + 83446410*x^12 + 19888027*x^11 - 390244377*x^10 + 110929757*x^9 + 994369765*x^8 - 511651057*x^7 - 1370318108*x^6 + 775297206*x^5 + 972267257*x^4 - 466392148*x^3 - 312747839*x^2 + 85444683*x + 38737459, 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]])