\\ Pari/GP code for working with number field 27.27.33967771186402966366427549984621879030979398934601324527479889.9 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^27 - 342*y^25 - 513*y^24 + 33345*y^23 + 97470*y^22 - 1260270*y^21 - 5221827*y^20 + 21068226*y^19 + 127768179*y^18 - 110794149*y^17 - 1625812596*y^16 - 1238190651*y^15 + 10625545845*y^14 + 21661040763*y^13 - 26176910337*y^12 - 123140488707*y^11 - 62085384675*y^10 + 266388984264*y^9 + 452485760760*y^8 + 50424782157*y^7 - 565685907975*y^6 - 671734642302*y^5 - 292929027750*y^4 + 19194328485*y^3 + 68592894498*y^2 + 24847653465*y + 2988651493, 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 - 342*x^25 - 513*x^24 + 33345*x^23 + 97470*x^22 - 1260270*x^21 - 5221827*x^20 + 21068226*x^19 + 127768179*x^18 - 110794149*x^17 - 1625812596*x^16 - 1238190651*x^15 + 10625545845*x^14 + 21661040763*x^13 - 26176910337*x^12 - 123140488707*x^11 - 62085384675*x^10 + 266388984264*x^9 + 452485760760*x^8 + 50424782157*x^7 - 565685907975*x^6 - 671734642302*x^5 - 292929027750*x^4 + 19194328485*x^3 + 68592894498*x^2 + 24847653465*x + 2988651493, 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]])