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 Γ a h , b h for h ≥ 3 with torsion group isomorphic to Z / 2 Z × Z / 4 Z such that Γ a h , b h 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 ( 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.