Some studies on noncongruency of an integer 10

Abstract

The only sense in which Tunnell``s theorem is not yet a completely satisfactory solution to the congruent number problem is that in one direction it is conditional upon the weak Birch-Swinnerton-Dyer conjecture for certain elliptic curves. So, we will use the fact that n is a congruent number if and only if the elliptic curve $E_n(Q)$ for $E_{n}:y^2 = x^3 - n^2x$ has nonzero rank r. By means of complete 2-descent, Chahal``s method and descent via two-isogeny, we get that the rank of the elliptic curve $E_{10}(Q)$ is 0. Hence we can conclude that 10 is not a congruent number.