We construct an elliptic curve of rank at least 6 over $Q(t)$ with three non-trivial rational points of order 2.
Shoichi Kihara. "On the rank of elliptic curves with three rational points of order 2. III." Proc. Japan Acad. Ser. A Math. Sci. 80 (3) 13 - 14, March 2004. https://doi.org/10.3792/pjaa.80.13