ELLIPTIC CURVES ARISING FROM THE TRIANGULAR NUMBERS.

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