\\ Pari/GP code for working with number field 23.23.294114910567428962511991419268717959339196150815159804184604281.1

\\ 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 - 330*y^21 + 491*y^20 + 46010*y^19 - 87998*y^18 - 3534734*y^17 + 8097381*y^16 + 163274386*y^15 - 432276476*y^14 - 4617725871*y^13 + 13958868541*y^12 + 76934101895*y^11 - 269759119969*y^10 - 654647210801*y^9 + 2901393808025*y^8 + 1396549312920*y^7 - 14112769866979*y^6 + 10842020004862*y^5 + 12214116095559*y^4 - 17293268074281*y^3 + 3381350907752*y^2 + 1669609496979*y - 220900980203, 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 - 330*x^21 + 491*x^20 + 46010*x^19 - 87998*x^18 - 3534734*x^17 + 8097381*x^16 + 163274386*x^15 - 432276476*x^14 - 4617725871*x^13 + 13958868541*x^12 + 76934101895*x^11 - 269759119969*x^10 - 654647210801*x^9 + 2901393808025*x^8 + 1396549312920*x^7 - 14112769866979*x^6 + 10842020004862*x^5 + 12214116095559*x^4 - 17293268074281*x^3 + 3381350907752*x^2 + 1669609496979*x - 220900980203, 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]])