\\ Pari/GP code for working with number field 23.23.434498173468775074445875189356350743903198920600715375079386035604631941932241.1

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

\\ Define the number field: 
K = bnfinit(y^23 - 4324*y^21 - 6486*y^20 + 5382299*y^19 - 23233933*y^18 - 2724444300*y^17 + 29686323629*y^16 + 512000507352*y^15 - 10294412120640*y^14 + 6885232378569*y^13 + 1102283075184770*y^12 - 8796561210816172*y^11 - 7798660667836453*y^10 + 474243077814357335*y^9 - 2826995282155771181*y^8 + 5949260040976823570*y^7 + 9167317157190582864*y^6 - 81864894718917833350*y^5 + 204445625295748936871*y^4 - 269173314235796280477*y^3 + 199912058984322799237*y^2 - 78929282232647458634*y + 12862216057817467245, 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 - 4324*x^21 - 6486*x^20 + 5382299*x^19 - 23233933*x^18 - 2724444300*x^17 + 29686323629*x^16 + 512000507352*x^15 - 10294412120640*x^14 + 6885232378569*x^13 + 1102283075184770*x^12 - 8796561210816172*x^11 - 7798660667836453*x^10 + 474243077814357335*x^9 - 2826995282155771181*x^8 + 5949260040976823570*x^7 + 9167317157190582864*x^6 - 81864894718917833350*x^5 + 204445625295748936871*x^4 - 269173314235796280477*x^3 + 199912058984322799237*x^2 - 78929282232647458634*x + 12862216057817467245, 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]])