\\ Pari/GP code for working with number field 44.0.191158492466532603992882058813386761401107535956984758058028173625115327713733756218489.3 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^44 - y^43 + 6*y^42 - 11*y^41 + 41*y^40 - 96*y^39 + 301*y^38 - 781*y^37 + 2286*y^36 - 6191*y^35 + 17621*y^34 - 48576*y^33 + 136681*y^32 - 379561*y^31 + 1062966*y^30 - 2960771*y^29 + 8275601*y^28 - 23079456*y^27 + 64457461*y^26 - 179854741*y^25 + 502142046*y^24 - 1401415751*y^23 + 3912125981*y^22 + 7007078755*y^21 + 12553551150*y^20 + 22481842625*y^19 + 40285913125*y^18 + 72123300000*y^17 + 129306265625*y^16 + 231310234375*y^15 + 415221093750*y^14 + 741330078125*y^13 + 1334775390625*y^12 + 2371875000000*y^11 + 4302001953125*y^10 + 7557373046875*y^9 + 13952636718750*y^8 + 23834228515625*y^7 + 45928955078125*y^6 + 73242187500000*y^5 + 156402587890625*y^4 + 209808349609375*y^3 + 572204589843750*y^2 + 476837158203125*y + 2384185791015625, 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^44 - x^43 + 6*x^42 - 11*x^41 + 41*x^40 - 96*x^39 + 301*x^38 - 781*x^37 + 2286*x^36 - 6191*x^35 + 17621*x^34 - 48576*x^33 + 136681*x^32 - 379561*x^31 + 1062966*x^30 - 2960771*x^29 + 8275601*x^28 - 23079456*x^27 + 64457461*x^26 - 179854741*x^25 + 502142046*x^24 - 1401415751*x^23 + 3912125981*x^22 + 7007078755*x^21 + 12553551150*x^20 + 22481842625*x^19 + 40285913125*x^18 + 72123300000*x^17 + 129306265625*x^16 + 231310234375*x^15 + 415221093750*x^14 + 741330078125*x^13 + 1334775390625*x^12 + 2371875000000*x^11 + 4302001953125*x^10 + 7557373046875*x^9 + 13952636718750*x^8 + 23834228515625*x^7 + 45928955078125*x^6 + 73242187500000*x^5 + 156402587890625*x^4 + 209808349609375*x^3 + 572204589843750*x^2 + 476837158203125*x + 2384185791015625, 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]])