\\ Pari/GP code for working with number field 32.32.8052845212573000012543979797231296934933304854055472857088000000000000000000000000.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^32 - 480*y^30 + 104400*y^28 - 13608000*y^26 + 1184625000*y^24 - 72657000000*y^22 + 3227647500000*y^20 - 105129090000000*y^18 + 2513242307812500*y^16 - 43708561875000000*y^14 + 542383517812500000*y^12 - 4649001581250000000*y^10 + 26150633894531250000*y^8 - 88933329843750000000*y^6 + 158809517578125000000*y^4 - 112100835937500000000*y^2 + 1226865178750781250, 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 - 480*x^30 + 104400*x^28 - 13608000*x^26 + 1184625000*x^24 - 72657000000*x^22 + 3227647500000*x^20 - 105129090000000*x^18 + 2513242307812500*x^16 - 43708561875000000*x^14 + 542383517812500000*x^12 - 4649001581250000000*x^10 + 26150633894531250000*x^8 - 88933329843750000000*x^6 + 158809517578125000000*x^4 - 112100835937500000000*x^2 + 1226865178750781250, 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]])