\\ Pari/GP code for working with number field 24.0.1729054511370401309967041015625.1

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

\\ Define the number field: 
K = bnfinit(y^24 - 8*y^23 + 35*y^22 - 105*y^21 + 233*y^20 - 377*y^19 + 373*y^18 + 86*y^17 - 1310*y^16 + 3226*y^15 - 4893*y^14 + 4424*y^13 - 205*y^12 - 6944*y^11 + 13153*y^10 - 14094*y^9 + 9116*y^8 - 2660*y^7 - 525*y^6 + 689*y^5 - 175*y^4 - 37*y^3 + 37*y^2 - 10*y + 1, 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 - 8*x^23 + 35*x^22 - 105*x^21 + 233*x^20 - 377*x^19 + 373*x^18 + 86*x^17 - 1310*x^16 + 3226*x^15 - 4893*x^14 + 4424*x^13 - 205*x^12 - 6944*x^11 + 13153*x^10 - 14094*x^9 + 9116*x^8 - 2660*x^7 - 525*x^6 + 689*x^5 - 175*x^4 - 37*x^3 + 37*x^2 - 10*x + 1, 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]])