Table of Complex Multiplication Moduli

A complete catalogue of all the numerical results in the solution of the niodnhiT equations for Complex Multiplication, when (he square of Hit! ratio of tho periods is an integer, or a rational fraction, is of course impossible; but a list of thosn cases which have been solved up till now will bo found useful ns a verification of the form and coellicients of l-hc modular equations, and also in some eases will serve for the determination of the coellicients when the form of the modular equation can be inferred from independent considerations.It is proposed, thru, to give here a list of these numerical results, us fiir us is known at present, embodying the values obtained by Abel, Jiinobi, Kroueoker, 1 Lcrmitc, .lonbert,and Weber, and more recently of K>.Russell and Muthews.The numerical results are also presented in more than one shape, for verification in case of a misprint.Denoting the ratio of the periods of the elliptic functions, K'/K, hy \/A, and taking A as infcegni.1,it is convenient to draw np the list in four classes, according to the form of A (ZVoc.Land.Math.Snr,., Vol.xix., p. 3U1) ; thus, in Class A, A E^ .' ?, mod.S; Glass 15, A s 7 , mod. S ; Class C, A = 1, mod.4; Class D, A ==2, mod.4; the class for A = 0, mod.4, not requiring much separate consideration.Class A : A s 3, mod.8.Here the simplest numerical result to tabnlato is Klein's absolute invariant J (Ma'k.Ann., Vol.xiv., p. J11), or Hcrmite's a (Equations modidaircs), connected with ./by the relation