\\ Pari/GP code for working with number field 30.2.114463058803512608608063235872919561806671142578125.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^30 - 3*y^29 + 9*y^28 + 14*y^27 - 113*y^26 + 370*y^25 - 643*y^24 - 15*y^23 + 3880*y^22 - 13302*y^21 + 28770*y^20 - 44390*y^19 + 53641*y^18 - 46319*y^17 + 35868*y^16 - 26390*y^15 + 12095*y^14 + 43878*y^13 + 45313*y^12 - 6914*y^11 - 59831*y^10 - 94496*y^9 - 64150*y^8 - 2932*y^7 + 54993*y^6 + 69070*y^5 + 52989*y^4 + 26576*y^3 + 8569*y^2 + 484*y - 121, 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 - 3*x^29 + 9*x^28 + 14*x^27 - 113*x^26 + 370*x^25 - 643*x^24 - 15*x^23 + 3880*x^22 - 13302*x^21 + 28770*x^20 - 44390*x^19 + 53641*x^18 - 46319*x^17 + 35868*x^16 - 26390*x^15 + 12095*x^14 + 43878*x^13 + 45313*x^12 - 6914*x^11 - 59831*x^10 - 94496*x^9 - 64150*x^8 - 2932*x^7 + 54993*x^6 + 69070*x^5 + 52989*x^4 + 26576*x^3 + 8569*x^2 + 484*x - 121, 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]])