\\ Pari/GP code for working with number field 24.4.641953627807088196277618408203125.4

\\ Some of these functions may take a long time to execute (this depends on the field).

\\ Define the number field: 
K = bnfinit(y^24 - 6*y^23 + 11*y^22 - y^21 - 42*y^20 + 132*y^19 - 157*y^18 - 93*y^17 + 463*y^16 - 504*y^15 + 89*y^14 + 756*y^13 - 1626*y^12 + 841*y^11 + 2572*y^10 - 3447*y^9 - 1348*y^8 + 4593*y^7 - 688*y^6 - 3121*y^5 + 1191*y^4 + 1049*y^3 - 514*y^2 - 151*y + 83, 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 - 6*x^23 + 11*x^22 - x^21 - 42*x^20 + 132*x^19 - 157*x^18 - 93*x^17 + 463*x^16 - 504*x^15 + 89*x^14 + 756*x^13 - 1626*x^12 + 841*x^11 + 2572*x^10 - 3447*x^9 - 1348*x^8 + 4593*x^7 - 688*x^6 - 3121*x^5 + 1191*x^4 + 1049*x^3 - 514*x^2 - 151*x + 83, 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]])