\\ Pari/GP code for working with number field 27.1.4907350451606845763597379498834801987276076711.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^27 - 3*y^26 + 15*y^25 - 53*y^24 + 106*y^23 - 298*y^22 + 714*y^21 - 199*y^20 + 2747*y^19 + 4180*y^18 + 4767*y^17 + 27487*y^16 - 7928*y^15 + 93863*y^14 - 80586*y^13 + 257840*y^12 - 253867*y^11 + 459460*y^10 - 363114*y^9 + 981160*y^8 + 143119*y^7 - 202978*y^6 + 143097*y^5 + 629777*y^4 + 25669*y^3 - 205103*y^2 - 65505*y + 40293, 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 - 3*x^26 + 15*x^25 - 53*x^24 + 106*x^23 - 298*x^22 + 714*x^21 - 199*x^20 + 2747*x^19 + 4180*x^18 + 4767*x^17 + 27487*x^16 - 7928*x^15 + 93863*x^14 - 80586*x^13 + 257840*x^12 - 253867*x^11 + 459460*x^10 - 363114*x^9 + 981160*x^8 + 143119*x^7 - 202978*x^6 + 143097*x^5 + 629777*x^4 + 25669*x^3 - 205103*x^2 - 65505*x + 40293, 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]])