\\ Pari/GP code for working with number field 27.27.13122249213311579236015222881822854848980675468493365969.5 \\ 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 - 144*y^25 + 1152*y^24 + 8748*y^23 - 53568*y^22 - 299961*y^21 + 1145826*y^20 + 5967756*y^19 - 11122377*y^18 - 64116414*y^17 + 40038543*y^16 + 341380242*y^15 - 21808332*y^14 - 928546470*y^13 - 170651646*y^12 + 1278782856*y^11 + 375962418*y^10 - 839220759*y^9 - 271274076*y^8 + 239523237*y^7 + 68849766*y^6 - 29543454*y^5 - 5065389*y^4 + 1859094*y^3 + 44577*y^2 - 39771*y + 2427, 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 - 144*x^25 + 1152*x^24 + 8748*x^23 - 53568*x^22 - 299961*x^21 + 1145826*x^20 + 5967756*x^19 - 11122377*x^18 - 64116414*x^17 + 40038543*x^16 + 341380242*x^15 - 21808332*x^14 - 928546470*x^13 - 170651646*x^12 + 1278782856*x^11 + 375962418*x^10 - 839220759*x^9 - 271274076*x^8 + 239523237*x^7 + 68849766*x^6 - 29543454*x^5 - 5065389*x^4 + 1859094*x^3 + 44577*x^2 - 39771*x + 2427, 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]])