Elliptic Curves With Congruence Polynomials And Rational Numbers On The Domain Of Rational Function

It is a long history for the problem of congruent number on integer. In this paper, we generalize the conception of classical congruent number to that polynomial and define the congruent polynomial in the function field k(t). We study the relationship between elliptic curve over function field and congruent polynomial. And get some result for non-congruent polynomial. In particular, it prove that when k=Fq with q=pf and p=3(mod4), n(t)= P1P2…Pl,statisfying degPi= 1(mod 2) with i=1,2 or degP1=1(mod2) with degPi=0(mod4)(2≤i≤l) and G(n(t)) being odd graph, n(t) is a non-congruent polynomial.