\\ Pari/GP code for working with number field 33.33.23681050358190252966666038984115482490423545779778728084912954382239302601.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^33 - 8*y^32 - 86*y^31 + 690*y^30 + 3527*y^29 - 25876*y^28 - 92162*y^27 + 549745*y^26 + 1676544*y^25 - 7221169*y^24 - 21518451*y^23 + 59984529*y^22 + 192193744*y^21 - 307177110*y^20 - 1170053321*y^19 + 864215343*y^18 + 4741699040*y^17 - 637264044*y^16 - 12404482875*y^15 - 3656463713*y^14 + 19997642137*y^13 + 12376322490*y^12 - 18399788371*y^11 - 16849661483*y^10 + 8289542908*y^9 + 11391691466*y^8 - 842642325*y^7 - 3707258605*y^6 - 522452271*y^5 + 463178377*y^4 + 128777001*y^3 - 2389174*y^2 - 1929880*y + 90889, 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^33 - 8*x^32 - 86*x^31 + 690*x^30 + 3527*x^29 - 25876*x^28 - 92162*x^27 + 549745*x^26 + 1676544*x^25 - 7221169*x^24 - 21518451*x^23 + 59984529*x^22 + 192193744*x^21 - 307177110*x^20 - 1170053321*x^19 + 864215343*x^18 + 4741699040*x^17 - 637264044*x^16 - 12404482875*x^15 - 3656463713*x^14 + 19997642137*x^13 + 12376322490*x^12 - 18399788371*x^11 - 16849661483*x^10 + 8289542908*x^9 + 11391691466*x^8 - 842642325*x^7 - 3707258605*x^6 - 522452271*x^5 + 463178377*x^4 + 128777001*x^3 - 2389174*x^2 - 1929880*x + 90889, 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]])