ELLIPTIC CURVES ARISING FROM THE TRIANGULAR NUMBERS

Abstract

We study the Legendre family of elliptic curves E_t : y ² = x(x - 1)(x - Δ _t ), parametrized by triangular numbers Δ _t = t(t + 1)/2. We prove that the rank of E_t over the function field $\stackrel{‒}{Q}\left(t\right)$ is 1, while the rank is 0 over $Q\left(t\right)$ . We also produce some infinite subfamilies whose Mordell-Weil rank is positive, and find high rank curves from within these families.