Constructing one-parameter families of elliptic curves with moderate rank Original Research Article

We give several new constructions for moderate rank elliptic curves over image. In particular we construct infinitely many rational elliptic surfaces (not in Weierstrass form) of rank 6 over image using polynomials of degree two in T. While our method generates linearly independent points, we are able to show the rank is exactly 6 without having to verify the points are independent. The method generalizes; however, the higher rank surfaces are not rational, and we need to check that the constructed points are linearly independent.