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In this paper, we present details of seven elliptic curves over$\mathbb{Q}(u)$ with rank $2$ and torsion group $\mathbb{Z}/ 8\mathbb{Z}$ andfive curves over $\mathbb{Q}(u)$ with rank $2$ and torsion group $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/ 6\mathbb{Z}$. We also exhibit some particularexamples of curves with high rank over $\mathbb{Q}$ by specialization of theparameter. We present several sets of infinitely many elliptic curves in bothtorsion groups and rank at least $3$ parametrized by elliptic curves havingpositive rank. In some of these sets we have performed calculations about thedistribution of the root number. This has relation with recent heuristicsconcerning the rank bound for elliptic curves by Park, Poonen, Voight and Wood.

Elliptic curves with torsion groups $\mathbb{Z}/8\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$