\\ Pari/GP code for working with number field 32.0.366225584701948244050176000000000000000000000000.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^32 - 3*y^30 + 4*y^28 - 9*y^26 + 27*y^24 - 93*y^22 + 188*y^20 - 279*y^18 + 581*y^16 - 1116*y^14 + 3008*y^12 - 5952*y^10 + 6912*y^8 - 9216*y^6 + 16384*y^4 - 49152*y^2 + 65536, 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^32 - 3*x^30 + 4*x^28 - 9*x^26 + 27*x^24 - 93*x^22 + 188*x^20 - 279*x^18 + 581*x^16 - 1116*x^14 + 3008*x^12 - 5952*x^10 + 6912*x^8 - 9216*x^6 + 16384*x^4 - 49152*x^2 + 65536, 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]])