\\ Pari/GP code for working with number field 24.4.270761008401829353605241639483649123576469719409942626953125.7.
\\ 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 + y^20 + 2320*y^19 - 17980*y^18 + 129920*y^17 - 684255*y^16 + 2876220*y^15 - 10654310*y^14 + 40280565*y^13 - 142954485*y^12 + 449059490*y^11 - 1294715585*y^10 + 2462674490*y^9 - 3946815610*y^8 + 457323765*y^7 + 7396883840*y^6 + 7993315965*y^5 - 45581880420*y^4 - 78522002395*y^3 + 103377044430*y^2 + 258196577105*y + 212661967555, 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 + x^20 + 2320*x^19 - 17980*x^18 + 129920*x^17 - 684255*x^16 + 2876220*x^15 - 10654310*x^14 + 40280565*x^13 - 142954485*x^12 + 449059490*x^11 - 1294715585*x^10 + 2462674490*x^9 - 3946815610*x^8 + 457323765*x^7 + 7396883840*x^6 + 7993315965*x^5 - 45581880420*x^4 - 78522002395*x^3 + 103377044430*x^2 + 258196577105*x + 212661967555, 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]])