\\ Pari/GP code for working with number field 27.9.656187196700948326335269302720488800256.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^27 - 9*y^26 + 27*y^25 - 24*y^24 - 9*y^23 - 63*y^22 + 273*y^21 + 144*y^20 - 2376*y^19 + 4050*y^18 - 396*y^17 - 3465*y^16 - 141*y^15 + 441*y^14 + 17019*y^13 - 33225*y^12 + 12852*y^11 + 17379*y^10 - 14277*y^9 + 1188*y^8 + 1143*y^7 - 501*y^6 + 891*y^5 + 216*y^4 + 129*y^3 + 18*y^2 + 9*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^27 - 9*x^26 + 27*x^25 - 24*x^24 - 9*x^23 - 63*x^22 + 273*x^21 + 144*x^20 - 2376*x^19 + 4050*x^18 - 396*x^17 - 3465*x^16 - 141*x^15 + 441*x^14 + 17019*x^13 - 33225*x^12 + 12852*x^11 + 17379*x^10 - 14277*x^9 + 1188*x^8 + 1143*x^7 - 501*x^6 + 891*x^5 + 216*x^4 + 129*x^3 + 18*x^2 + 9*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]])