\\ Pari/GP code for working with number field 24.0.7404154726819150028835278894923776.1

\\ 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 + 8*y^22 - 12*y^21 + 12*y^20 - 4*y^19 - 8*y^18 + 24*y^17 - 56*y^16 + 88*y^15 - 88*y^14 + 56*y^13 - 28*y^12 + 112*y^11 - 352*y^10 + 704*y^9 - 896*y^8 + 768*y^7 - 512*y^6 - 512*y^5 + 3072*y^4 - 6144*y^3 + 8192*y^2 - 8192*y + 4096, 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

\\ 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 + 8*x^22 - 12*x^21 + 12*x^20 - 4*x^19 - 8*x^18 + 24*x^17 - 56*x^16 + 88*x^15 - 88*x^14 + 56*x^13 - 28*x^12 + 112*x^11 - 352*x^10 + 704*x^9 - 896*x^8 + 768*x^7 - 512*x^6 - 512*x^5 + 3072*x^4 - 6144*x^3 + 8192*x^2 - 8192*x + 4096, 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(b)]

\\ Galois group: 
polgalois(K.pol)

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