\\ Pari/GP code for working with number field 24.0.368947264000000000000000000.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^24 - 6*y^23 + 21*y^22 - 54*y^21 + 107*y^20 - 178*y^19 + 265*y^18 - 376*y^17 + 537*y^16 - 744*y^15 + 966*y^14 - 1140*y^13 + 1221*y^12 - 1228*y^11 + 1176*y^10 - 1074*y^9 + 914*y^8 - 696*y^7 + 471*y^6 - 280*y^5 + 140*y^4 - 58*y^3 + 21*y^2 - 6*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 - 6*x^23 + 21*x^22 - 54*x^21 + 107*x^20 - 178*x^19 + 265*x^18 - 376*x^17 + 537*x^16 - 744*x^15 + 966*x^14 - 1140*x^13 + 1221*x^12 - 1228*x^11 + 1176*x^10 - 1074*x^9 + 914*x^8 - 696*x^7 + 471*x^6 - 280*x^5 + 140*x^4 - 58*x^3 + 21*x^2 - 6*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]])