\\ Pari/GP code for working with number field 28.2.5576015581167650909427113418703236578781943428571.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^28 - 20*y^26 + 294*y^24 - 3361*y^22 + 28147*y^20 - 154673*y^18 + 536790*y^16 - 1083403*y^14 + 980940*y^12 + 64462*y^10 - 677733*y^8 - 44521*y^6 + 115444*y^4 + 5415*y^2 - 6859, 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^28 - 20*x^26 + 294*x^24 - 3361*x^22 + 28147*x^20 - 154673*x^18 + 536790*x^16 - 1083403*x^14 + 980940*x^12 + 64462*x^10 - 677733*x^8 - 44521*x^6 + 115444*x^4 + 5415*x^2 - 6859, 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]])