\\ Pari/GP code for working with number field 32.0.515207889456069254383620937908064472169867121.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^32 - 5*y^31 + 22*y^30 - 73*y^29 + 220*y^28 - 519*y^27 + 1255*y^26 - 2247*y^25 + 4413*y^24 - 6238*y^23 + 9744*y^22 - 10830*y^21 + 13240*y^20 - 10628*y^19 + 12053*y^18 - 5501*y^17 + 12591*y^16 - 3062*y^15 + 17256*y^14 + 194*y^13 + 18165*y^12 + 5659*y^11 + 13232*y^10 - 2318*y^9 - 319*y^8 - 13445*y^7 - 6940*y^6 - 4990*y^5 + 1466*y^4 + 2074*y^3 + 1626*y^2 + 264*y + 79, 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 - 5*x^31 + 22*x^30 - 73*x^29 + 220*x^28 - 519*x^27 + 1255*x^26 - 2247*x^25 + 4413*x^24 - 6238*x^23 + 9744*x^22 - 10830*x^21 + 13240*x^20 - 10628*x^19 + 12053*x^18 - 5501*x^17 + 12591*x^16 - 3062*x^15 + 17256*x^14 + 194*x^13 + 18165*x^12 + 5659*x^11 + 13232*x^10 - 2318*x^9 - 319*x^8 - 13445*x^7 - 6940*x^6 - 4990*x^5 + 1466*x^4 + 2074*x^3 + 1626*x^2 + 264*x + 79, 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]])