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㇍i=0UV(x i)UR2Q(t)[x]:;(2.1) Then,wrehave PNPtʹ(x)UR=Qt(x)2j Ft(x);whereQt|ristheuniquemonicpSolynomialdenedorverQ(t)sucrhthatFthasdegree4.ThediscriminanrtofthepSolynomialFtʹ(x)isnonzerofortUR2QnS ,where (color push Black color pop l$S)=URf1;2;4; 5; 6; 8; 11g.Hence,forsucrhparameterst,thequarticellipticcurve html: html: B:Et:yn92=URFtʹ(x);(2.2)conrtain0{thepSointsPi1=+(i;)Qtʹ(i))fori=0;:::ʜ;)9whicrhformanarithmeticprogressionoflength10onthecurvreEtʹ.FE c ffx ff ̟ ff ̎̄ c ffhtml: html:3(Thezsequenceoflength12b#ThereK2doSesn'tseemtobeawrayK2toconstructaninnitefamilyofquarticswithanarithmeticprogressionXoflength12fromfamilyofcurvres(html:color push cmyk 0 1 0 02.2 html: color pop).
Inthissectionweshallconstructanotherq