Elliptic curves and triangles with three rational medians
In [\textit{R. H. Buchholz}, J. Number Theory 97, No. 1, 113--131 (2002; Zbl 1037.11020)], it is proved that every rational sided triangle with three rational medians corresponds to a points on a one-parameter family of elliptic curves \(E[m]:y^2=x^3+9(1+m)^2x^2-576(m^2-1)^2m^2x\) and the rank of \(E[m]\) over \(\mathbb Q(m)\) is at least \(2\). In addition, Buchholz states that high rank curves with rank \(5,6,7\) emerge among \(E[m]~(m\in\mathbb Q)\). In this article, the authors prove that the rank of the elliptic curve \(E[m]\) over \(\mathbb Q(m)\) is exactly \(3\). By choosing \(m=(u^2-9)/(6u-73)\) or \(m=-(u^2-9)/4u\) they exhibit subfamilies of elliptic curves whose rank at least \(4\) over \(\mathbb Q(u)\). Further they prove the existence of infinitely many curves of rank at least \(5\) over \(\mathbb Q\) parameterized by an elliptic curve of positive rank and show examples of curves within those families having rank \(9\) and \(10\). These results are obtained from constructing independent points explicitly.
scientific article; zbMATH DE number 6154108