\\ Pari/GP code for working with number field 45.45.23518854458506786548183561086555508726667593641132303121759124944650820067393315326853553415276110172271728515625.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^45 - 15*y^44 - 90*y^43 + 2245*y^42 + 840*y^41 - 149682*y^40 + 233625*y^39 + 5893830*y^38 - 15309120*y^37 - 153159405*y^36 + 505833687*y^35 + 2780167860*y^34 - 10777640365*y^33 - 36390969195*y^32 + 160894854450*y^31 + 349558305696*y^30 - 1751940770010*y^29 - 2483116182570*y^28 + 14226276458010*y^27 + 13037929556940*y^26 - 87229174471284*y^25 - 50137706272620*y^24 + 406336593221610*y^23 + 138267603359730*y^22 - 1439822630956295*y^21 - 262195938148443*y^20 + 3868718864967360*y^19 + 311910060390275*y^18 - 7821266721741750*y^17 - 175081417856730*y^16 + 11738415157749574*y^15 - 32411856380685*y^14 - 12806776910595090*y^13 + 44301971232950*y^12 + 9836514943009755*y^11 + 117791800035840*y^10 - 5063789878500180*y^9 - 176503490451285*y^8 + 1618208036238660*y^7 + 90536277582715*y^6 - 283553373081756*y^5 - 18640864878495*y^4 + 21751317924475*y^3 + 974579665770*y^2 - 434206226325*y + 15265738099, 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^45 - 15*x^44 - 90*x^43 + 2245*x^42 + 840*x^41 - 149682*x^40 + 233625*x^39 + 5893830*x^38 - 15309120*x^37 - 153159405*x^36 + 505833687*x^35 + 2780167860*x^34 - 10777640365*x^33 - 36390969195*x^32 + 160894854450*x^31 + 349558305696*x^30 - 1751940770010*x^29 - 2483116182570*x^28 + 14226276458010*x^27 + 13037929556940*x^26 - 87229174471284*x^25 - 50137706272620*x^24 + 406336593221610*x^23 + 138267603359730*x^22 - 1439822630956295*x^21 - 262195938148443*x^20 + 3868718864967360*x^19 + 311910060390275*x^18 - 7821266721741750*x^17 - 175081417856730*x^16 + 11738415157749574*x^15 - 32411856380685*x^14 - 12806776910595090*x^13 + 44301971232950*x^12 + 9836514943009755*x^11 + 117791800035840*x^10 - 5063789878500180*x^9 - 176503490451285*x^8 + 1618208036238660*x^7 + 90536277582715*x^6 - 283553373081756*x^5 - 18640864878495*x^4 + 21751317924475*x^3 + 974579665770*x^2 - 434206226325*x + 15265738099, 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]])