On the rank of the elliptic curves with a rational point of order 4

Abstract

We construct an elliptic curve of rank at least 4 over $Q(t)$ with a rational point of order 4. We also show an infinite family of elliptic curves of rank at least 5 over $Q$ with a rational point of order4, which is parametrized by the rational points of an elliptic curve of rank at least 1.