Pairing powers of pythagorean pairs

A pair (a, b) of positive integers is a pythagorean pair if a² + b² 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 (aʰk, bʰl) is a pythagorean pair. To each pythagorean pair (a, b) we assign an elliptic curve Γ_(aʰ,bʰ) for h ≥ 3 with torsion group isomorphic to Z/2Z x Z/4Z such that Γ_(aʰ,bʰ) has positive rank over 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 (aʰk, bʰl) 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.