\\ Pari/GP code for working with number field 18.0.39655567350356468482252381576612622224820465664.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^18 - 3*y^17 + 186*y^16 - 796*y^15 + 20886*y^14 - 43086*y^13 + 1499866*y^12 - 4087764*y^11 + 87804711*y^10 - 169457551*y^9 + 3935286084*y^8 - 1528031184*y^7 + 87470587978*y^6 + 226709260608*y^5 + 1850764572768*y^4 + 12656412331040*y^3 + 48056367294720*y^2 + 195986267080704*y + 353454875222016, 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^18 - 3*x^17 + 186*x^16 - 796*x^15 + 20886*x^14 - 43086*x^13 + 1499866*x^12 - 4087764*x^11 + 87804711*x^10 - 169457551*x^9 + 3935286084*x^8 - 1528031184*x^7 + 87470587978*x^6 + 226709260608*x^5 + 1850764572768*x^4 + 12656412331040*x^3 + 48056367294720*x^2 + 195986267080704*x + 353454875222016, 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]])