\\ Pari/GP code for working with number field 24.4.6769025210045733840131040987091228089411742985248565673828125.6.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^24 - 2*y^23 + 119*y^22 - 123*y^21 - 2284*y^20 + 35021*y^19 - 287932*y^18 - 3528451*y^17 + 430787*y^16 + 76267881*y^15 + 383091958*y^14 + 1547361734*y^13 + 1926262502*y^12 - 24636617534*y^11 - 181070920997*y^10 - 710772344063*y^9 - 1879981260494*y^8 - 2702189457572*y^7 + 3316968239109*y^6 + 29845357531527*y^5 + 75565793089194*y^4 + 102052357360777*y^3 + 77691878905336*y^2 + 30942834489368*y + 4794247932259, 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

\\ Narrow class group: 
bnfnarrow(K)

\\ 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^24 - 2*x^23 + 119*x^22 - 123*x^21 - 2284*x^20 + 35021*x^19 - 287932*x^18 - 3528451*x^17 + 430787*x^16 + 76267881*x^15 + 383091958*x^14 + 1547361734*x^13 + 1926262502*x^12 - 24636617534*x^11 - 181070920997*x^10 - 710772344063*x^9 - 1879981260494*x^8 - 2702189457572*x^7 + 3316968239109*x^6 + 29845357531527*x^5 + 75565793089194*x^4 + 102052357360777*x^3 + 77691878905336*x^2 + 30942834489368*x + 4794247932259, 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(L)]

\\ Galois group: 
polgalois(K.pol)

\\ Frobenius cycle types: 
\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])