\\ Pari/GP code for working with number field 23.23.477949960252640343082669666642744217085836889062320413931701625489.1

\\ 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 - 462*y^21 + 519*y^20 + 78046*y^19 - 96292*y^18 - 6163501*y^17 + 7373917*y^16 + 250455980*y^15 - 290053856*y^14 - 5501340108*y^13 + 6875865393*y^12 + 66926124686*y^11 - 96263348909*y^10 - 438046106473*y^9 + 740824871406*y^8 + 1383038185176*y^7 - 2863260767644*y^6 - 1570567685623*y^5 + 5193710248505*y^4 - 862947136357*y^3 - 3368621429671*y^2 + 2154713129200*y - 341763031283, 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 - 462*x^21 + 519*x^20 + 78046*x^19 - 96292*x^18 - 6163501*x^17 + 7373917*x^16 + 250455980*x^15 - 290053856*x^14 - 5501340108*x^13 + 6875865393*x^12 + 66926124686*x^11 - 96263348909*x^10 - 438046106473*x^9 + 740824871406*x^8 + 1383038185176*x^7 - 2863260767644*x^6 - 1570567685623*x^5 + 5193710248505*x^4 - 862947136357*x^3 - 3368621429671*x^2 + 2154713129200*x - 341763031283, 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]])