\\ Pari/GP code for working with number field 24.4.1211192363043189952963494621037336955654250561565277166664600372314453125.8.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^24 - 12*y^23 - 40*y^22 + 401*y^21 - 930*y^20 + 247873*y^19 - 1200746*y^18 - 6559960*y^17 - 64580422*y^16 - 2923221605*y^15 - 12334220651*y^14 - 93433926183*y^13 - 127075389375*y^12 + 9477419064504*y^11 + 83412644565090*y^10 + 530594726847237*y^9 + 3434957990181416*y^8 + 5105977496091715*y^7 - 45579398797207878*y^6 - 385128811285350640*y^5 + 2216757763306056151*y^4 - 22438363614405112672*y^3 + 141472039110471732990*y^2 - 220230876669439407034*y + 777019087232248674575, 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 - 12*x^23 - 40*x^22 + 401*x^21 - 930*x^20 + 247873*x^19 - 1200746*x^18 - 6559960*x^17 - 64580422*x^16 - 2923221605*x^15 - 12334220651*x^14 - 93433926183*x^13 - 127075389375*x^12 + 9477419064504*x^11 + 83412644565090*x^10 + 530594726847237*x^9 + 3434957990181416*x^8 + 5105977496091715*x^7 - 45579398797207878*x^6 - 385128811285350640*x^5 + 2216757763306056151*x^4 - 22438363614405112672*x^3 + 141472039110471732990*x^2 - 220230876669439407034*x + 777019087232248674575, 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]])