\\ Pari/GP code for working with number field 30.2.301337616246914973122131519082482771026611328125.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^30 - 5*y^29 + 8*y^28 + 19*y^27 - 63*y^26 - 52*y^25 + 265*y^24 - 211*y^23 - 393*y^22 + 858*y^21 - 86*y^20 - 2380*y^19 + 4186*y^18 + 8272*y^17 - 3336*y^16 - 13036*y^15 + 7176*y^14 + 23259*y^13 + 6583*y^12 - 9115*y^11 - 7328*y^10 + 1279*y^9 + 4440*y^8 + 2429*y^7 - 349*y^6 - 1189*y^5 - 88*y^4 + 271*y^3 + 100*y^2 - 13*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^30 - 5*x^29 + 8*x^28 + 19*x^27 - 63*x^26 - 52*x^25 + 265*x^24 - 211*x^23 - 393*x^22 + 858*x^21 - 86*x^20 - 2380*x^19 + 4186*x^18 + 8272*x^17 - 3336*x^16 - 13036*x^15 + 7176*x^14 + 23259*x^13 + 6583*x^12 - 9115*x^11 - 7328*x^10 + 1279*x^9 + 4440*x^8 + 2429*x^7 - 349*x^6 - 1189*x^5 - 88*x^4 + 271*x^3 + 100*x^2 - 13*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]])