\\ Pari/GP code for working with number field 42.0.213117637842219136349783160187126852219651508028407629359584264706447783442638664863363.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^42 - 4*y^41 + 62*y^40 - 200*y^39 + 2132*y^38 - 6208*y^37 + 46584*y^36 - 115568*y^35 + 701352*y^34 - 1537594*y^33 + 7769821*y^32 - 14875116*y^31 + 64656220*y^30 - 109927490*y^29 + 414171141*y^28 - 618138014*y^27 + 2041345075*y^26 - 2683015728*y^25 + 7816607585*y^24 - 8902561137*y^23 + 22976431713*y^22 - 22760113866*y^21 + 52536783259*y^20 - 45434427911*y^19 + 92226880868*y^18 - 71336683352*y^17 + 125874997329*y^16 - 89801104142*y^15 + 131144820279*y^14 - 89289000509*y^13 + 106600416700*y^12 - 67935846970*y^11 + 65774842016*y^10 - 38731708226*y^9 + 30551737937*y^8 - 15767508635*y^7 + 9878705051*y^6 - 4297296085*y^5 + 2147704850*y^4 - 731044046*y^3 + 251922248*y^2 - 47762400*y + 8082649, 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^42 - 4*x^41 + 62*x^40 - 200*x^39 + 2132*x^38 - 6208*x^37 + 46584*x^36 - 115568*x^35 + 701352*x^34 - 1537594*x^33 + 7769821*x^32 - 14875116*x^31 + 64656220*x^30 - 109927490*x^29 + 414171141*x^28 - 618138014*x^27 + 2041345075*x^26 - 2683015728*x^25 + 7816607585*x^24 - 8902561137*x^23 + 22976431713*x^22 - 22760113866*x^21 + 52536783259*x^20 - 45434427911*x^19 + 92226880868*x^18 - 71336683352*x^17 + 125874997329*x^16 - 89801104142*x^15 + 131144820279*x^14 - 89289000509*x^13 + 106600416700*x^12 - 67935846970*x^11 + 65774842016*x^10 - 38731708226*x^9 + 30551737937*x^8 - 15767508635*x^7 + 9878705051*x^6 - 4297296085*x^5 + 2147704850*x^4 - 731044046*x^3 + 251922248*x^2 - 47762400*x + 8082649, 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]])