\\ Pari/GP code for working with number field 24.4.270761008401829353605241639483649123576469719409942626953125.2.
\\ 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 - 1304*y^20 + 2465*y^19 + 2465*y^18 - 1885*y^17 - 599575*y^16 + 5496080*y^15 + 20488210*y^14 - 25107040*y^13 - 597431900*y^12 - 567629035*y^11 + 3603058165*y^10 + 9448143015*y^9 - 17099510555*y^8 + 16736471880*y^7 + 402014550880*y^6 + 156208942105*y^5 - 3470676872911*y^4 - 7274032113936*y^3 + 8069786480454*y^2 + 22227043889439*y - 6585670043891, 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 - 1304*x^20 + 2465*x^19 + 2465*x^18 - 1885*x^17 - 599575*x^16 + 5496080*x^15 + 20488210*x^14 - 25107040*x^13 - 597431900*x^12 - 567629035*x^11 + 3603058165*x^10 + 9448143015*x^9 - 17099510555*x^8 + 16736471880*x^7 + 402014550880*x^6 + 156208942105*x^5 - 3470676872911*x^4 - 7274032113936*x^3 + 8069786480454*x^2 + 22227043889439*x - 6585670043891, 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]])