\\ Pari/GP code for working with number field 17.17.462899178676311160288470191817767102492054133121.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^17 - y^16 - 448*y^15 + 1309*y^14 + 75494*y^13 - 374314*y^12 - 5597667*y^11 + 41830550*y^10 + 136426018*y^9 - 1919481097*y^8 + 2548312782*y^7 + 27117309376*y^6 - 105628969954*y^5 + 34101907629*y^4 + 457076113374*y^3 - 817186035962*y^2 + 323264486326*y + 117028501127, 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^17 - x^16 - 448*x^15 + 1309*x^14 + 75494*x^13 - 374314*x^12 - 5597667*x^11 + 41830550*x^10 + 136426018*x^9 - 1919481097*x^8 + 2548312782*x^7 + 27117309376*x^6 - 105628969954*x^5 + 34101907629*x^4 + 457076113374*x^3 - 817186035962*x^2 + 323264486326*x + 117028501127, 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]])