\\ Pari/GP code for working with number field 24.4.49177850545349555386638457176064.2

\\ 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 + 7*y^22 - 10*y^21 + 23*y^20 - 32*y^19 - 21*y^18 + 156*y^17 - 359*y^16 + 654*y^15 - 876*y^14 + 646*y^13 + 61*y^12 - 1012*y^11 + 2338*y^10 - 3178*y^9 + 2897*y^8 - 2474*y^7 + 1335*y^6 - 52*y^5 - 44*y^4 + 454*y^3 - 524*y^2 + 62*y - 197, 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 + 7*x^22 - 10*x^21 + 23*x^20 - 32*x^19 - 21*x^18 + 156*x^17 - 359*x^16 + 654*x^15 - 876*x^14 + 646*x^13 + 61*x^12 - 1012*x^11 + 2338*x^10 - 3178*x^9 + 2897*x^8 - 2474*x^7 + 1335*x^6 - 52*x^5 - 44*x^4 + 454*x^3 - 524*x^2 + 62*x - 197, 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]])