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^hk,b^hl) 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 \mathbb Z/2\mathbb Z \times \mathbb Z/4\mathbb 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^hk,b^hl) 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.
