\\ Pari/GP code for working with number field 27.27.13122249213311579236015222881822854848980675468493365969.2 \\ 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 + 1314*y^24 + 7740*y^23 - 68076*y^22 - 239166*y^21 + 1714032*y^20 + 4990779*y^19 - 22424565*y^18 - 67949397*y^17 + 139167009*y^16 + 521720982*y^15 - 259125408*y^14 - 1876452975*y^13 - 322111764*y^12 + 3304120158*y^11 + 1504663875*y^10 - 3180640551*y^9 - 1866753171*y^8 + 1770577101*y^7 + 1116671832*y^6 - 573082407*y^5 - 338304879*y^4 + 102031929*y^3 + 44766450*y^2 - 8037450*y - 1183123, 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 + 1314*x^24 + 7740*x^23 - 68076*x^22 - 239166*x^21 + 1714032*x^20 + 4990779*x^19 - 22424565*x^18 - 67949397*x^17 + 139167009*x^16 + 521720982*x^15 - 259125408*x^14 - 1876452975*x^13 - 322111764*x^12 + 3304120158*x^11 + 1504663875*x^10 - 3180640551*x^9 - 1866753171*x^8 + 1770577101*x^7 + 1116671832*x^6 - 573082407*x^5 - 338304879*x^4 + 102031929*x^3 + 44766450*x^2 - 8037450*x - 1183123, 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]])