\\ Pari/GP code for working with number field 23.23.16151009482177927765006229664562804196212236775629575713829087641.1

\\ 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 - 396*y^21 + 517*y^20 + 58242*y^19 - 89048*y^18 - 4136665*y^17 + 6675371*y^16 + 155317993*y^15 - 248917597*y^14 - 3168505860*y^13 + 4803054368*y^12 + 34789513687*y^11 - 47690452504*y^10 - 193631555469*y^9 + 225956060874*y^8 + 483028561517*y^7 - 415966978205*y^6 - 465412635150*y^5 + 267147031000*y^4 + 115014666875*y^3 - 62643096875*y^2 - 1127109375*y + 2017578125, 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 - 396*x^21 + 517*x^20 + 58242*x^19 - 89048*x^18 - 4136665*x^17 + 6675371*x^16 + 155317993*x^15 - 248917597*x^14 - 3168505860*x^13 + 4803054368*x^12 + 34789513687*x^11 - 47690452504*x^10 - 193631555469*x^9 + 225956060874*x^8 + 483028561517*x^7 - 415966978205*x^6 - 465412635150*x^5 + 267147031000*x^4 + 115014666875*x^3 - 62643096875*x^2 - 1127109375*x + 2017578125, 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]])