\\ Pari/GP code for working with number field 23.23.824185149135487077883465900577270766354751717380230010246241.1

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

\\ Define the number field: 
K = bnfinit(y^23 - 253*y^21 - 161*y^20 + 26335*y^19 + 33028*y^18 - 1467515*y^17 - 2734079*y^16 + 47658093*y^15 + 118763283*y^14 - 910024555*y^13 - 2931964974*y^12 + 9516128606*y^11 + 41347298260*y^10 - 39738537224*y^9 - 313669878607*y^8 - 104709853475*y^7 + 1076728692839*y^6 + 1230914947669*y^5 - 913297744524*y^4 - 1823221987393*y^3 - 124828056170*y^2 + 645413252986*y + 157112485811, 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 - 253*x^21 - 161*x^20 + 26335*x^19 + 33028*x^18 - 1467515*x^17 - 2734079*x^16 + 47658093*x^15 + 118763283*x^14 - 910024555*x^13 - 2931964974*x^12 + 9516128606*x^11 + 41347298260*x^10 - 39738537224*x^9 - 313669878607*x^8 - 104709853475*x^7 + 1076728692839*x^6 + 1230914947669*x^5 - 913297744524*x^4 - 1823221987393*x^3 - 124828056170*x^2 + 645413252986*x + 157112485811, 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]])