\\ Pari/GP code for working with number field 23.23.12687910093209570367523581223700258577242701778943851443146801.1

\\ 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 - 286*y^21 + 61*y^20 + 32489*y^19 + 14967*y^18 - 1928206*y^17 - 2239956*y^16 + 65439786*y^15 + 128285990*y^14 - 1279328514*y^13 - 3758213062*y^12 + 12892455511*y^11 + 58719404143*y^10 - 31105194350*y^9 - 448954503771*y^8 - 491325793721*y^7 + 1084631540509*y^6 + 3053149451318*y^5 + 2305895458302*y^4 - 281144721703*y^3 - 1007901237378*y^2 - 154197339893*y + 118359867167, 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 - 286*x^21 + 61*x^20 + 32489*x^19 + 14967*x^18 - 1928206*x^17 - 2239956*x^16 + 65439786*x^15 + 128285990*x^14 - 1279328514*x^13 - 3758213062*x^12 + 12892455511*x^11 + 58719404143*x^10 - 31105194350*x^9 - 448954503771*x^8 - 491325793721*x^7 + 1084631540509*x^6 + 3053149451318*x^5 + 2305895458302*x^4 - 281144721703*x^3 - 1007901237378*x^2 - 154197339893*x + 118359867167, 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]])