\\ Pari/GP code for working with number field 19.3.146525077331272123613812972.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^19 - 2*y^18 + 7*y^16 - 10*y^15 - y^14 + 19*y^13 - 19*y^12 - 9*y^11 + 33*y^10 - 19*y^9 - 20*y^8 + 32*y^7 - 9*y^6 - 17*y^5 + 17*y^4 - 3*y^3 - 5*y^2 + 4*y - 1, 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^19 - 2*x^18 + 7*x^16 - 10*x^15 - x^14 + 19*x^13 - 19*x^12 - 9*x^11 + 33*x^10 - 19*x^9 - 20*x^8 + 32*x^7 - 9*x^6 - 17*x^5 + 17*x^4 - 3*x^3 - 5*x^2 + 4*x - 1, 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]])