\\ Pari/GP code for working with number field 24.4.2295251366305746200663037028252313182580209076404571533203125.4.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^24 - 4*y^23 + 6*y^22 - 4*y^21 + 446*y^20 + 12994*y^19 + 25454*y^18 + 45034*y^17 - 18156*y^16 - 12083886*y^15 + 6061701*y^14 + 665735486*y^13 + 2209626941*y^12 - 7709179644*y^11 - 98459485684*y^10 - 168718807126*y^9 + 2406563134669*y^8 + 13899669932829*y^7 + 14257456475664*y^6 - 90366861809451*y^5 - 326471613749334*y^4 - 443687608352359*y^3 - 313677448742334*y^2 - 119801317951219*y - 8058266421349, 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 - 4*x^23 + 6*x^22 - 4*x^21 + 446*x^20 + 12994*x^19 + 25454*x^18 + 45034*x^17 - 18156*x^16 - 12083886*x^15 + 6061701*x^14 + 665735486*x^13 + 2209626941*x^12 - 7709179644*x^11 - 98459485684*x^10 - 168718807126*x^9 + 2406563134669*x^8 + 13899669932829*x^7 + 14257456475664*x^6 - 90366861809451*x^5 - 326471613749334*x^4 - 443687608352359*x^3 - 313677448742334*x^2 - 119801317951219*x - 8058266421349, 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]])