\\ Pari/GP code for working with number field 27.9.67585198634817523235520443624317923.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^27 - 18*y^23 - 18*y^22 - 18*y^21 + 18*y^20 + 135*y^19 + 219*y^18 - 162*y^17 + 324*y^16 - 153*y^15 - 81*y^14 - 675*y^13 - 630*y^12 - 540*y^11 - 513*y^10 - 405*y^9 + 27*y^8 + 243*y^7 + 387*y^6 + 162*y^5 - 81*y^4 - 135*y^3 + 27*y - 3, 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 - 18*x^23 - 18*x^22 - 18*x^21 + 18*x^20 + 135*x^19 + 219*x^18 - 162*x^17 + 324*x^16 - 153*x^15 - 81*x^14 - 675*x^13 - 630*x^12 - 540*x^11 - 513*x^10 - 405*x^9 + 27*x^8 + 243*x^7 + 387*x^6 + 162*x^5 - 81*x^4 - 135*x^3 + 27*x - 3, 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]])