Pairing Powers of Pythagorean Pairs

A pair $(a,b)$ of positive integers is a pythagorean pair if $a^2+b^2$ is a square. A pythagorean pair $(a,b)$ is called a pythapotent pair of degree $h$ if there is another pythagorean pair $(k,l)$, which is not a multiple of $(a,b)$, such that $(a^{h}k,b^{h}l)$ is a pythagorean pair. To each pythagorean pair $(a,b)$ we assign an elliptic curve ${\mathrm{\Gamma }}_{a^h,b^h}$ for $h\ge 3$ with torsion group isomorphic to $\mathbb{Z}\mathrm{/}2\mathbb{Z}\times \mathbb{Z}\mathrm{/}4\mathbb{Z}$ such that ${\mathrm{\Gamma }}_{a^h,b^h}$ has positive rank over $\mathbb{Q}$ if and only if $(a,b)$ is a pythapotent pair of degree $h$. As a side result, we get that if $(a,b)$ is a pythapotent pair of degree $h$, then there exist infinitely many pythagorean pairs $(k,l)$, not multiples of each other, such that $(a^{h}k,b^{h}l)$ is a pythagorean pair. In particular, we show that any pythagorean pair is always a pythapotent pair of degree 3. In a previous work, pythapotent pairs of degrees 1 and 2 have been studied.