\\ Pari/GP code for working with number field 27.1.8138911451501750747538217172562287688025999.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^27 - 11*y^26 + 70*y^25 - 273*y^24 + 723*y^23 - 1456*y^22 + 2649*y^21 - 4775*y^20 + 8022*y^19 - 11719*y^18 + 15552*y^17 - 20687*y^16 + 27099*y^15 - 31222*y^14 + 31020*y^13 - 30638*y^12 + 32802*y^11 - 31588*y^10 + 22446*y^9 - 12521*y^8 + 9384*y^7 - 8740*y^6 + 4644*y^5 - 254*y^4 - 460*y^3 - 287*y^2 + 245*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 - 11*x^26 + 70*x^25 - 273*x^24 + 723*x^23 - 1456*x^22 + 2649*x^21 - 4775*x^20 + 8022*x^19 - 11719*x^18 + 15552*x^17 - 20687*x^16 + 27099*x^15 - 31222*x^14 + 31020*x^13 - 30638*x^12 + 32802*x^11 - 31588*x^10 + 22446*x^9 - 12521*x^8 + 9384*x^7 - 8740*x^6 + 4644*x^5 - 254*x^4 - 460*x^3 - 287*x^2 + 245*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]])