\\ Pari/GP code for working with number field 20.2.558854584859141340505879659895771.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^20 - 8*y^19 + 38*y^18 - 133*y^17 + 323*y^16 - 570*y^15 + 608*y^14 + 608*y^13 - 2280*y^12 + 5662*y^11 - 2527*y^10 + 4731*y^9 + 8265*y^8 - 6935*y^7 + 27265*y^6 - 20539*y^5 + 29469*y^4 - 20178*y^3 + 16226*y^2 - 6696*y + 729, 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^20 - 8*x^19 + 38*x^18 - 133*x^17 + 323*x^16 - 570*x^15 + 608*x^14 + 608*x^13 - 2280*x^12 + 5662*x^11 - 2527*x^10 + 4731*x^9 + 8265*x^8 - 6935*x^7 + 27265*x^6 - 20539*x^5 + 29469*x^4 - 20178*x^3 + 16226*x^2 - 6696*x + 729, 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]])