  We study the family of elliptic curves $y^2=x(x-a^2)(x-b^2)$ parametrized by
Pythagorean triples $(a,b,c)$. We prove that for a generic triple the lower
bound of the rank of the Mordell-Weil group over $\mathbb{Q}$ is 1, and for
some explicitly given infinite family the rank is 2. To each family we attach
an elliptic surface fibered over the projective line. We show that the lower
bounds for the rank are optimal, in the sense that for each generic fiber of
such an elliptic surface its corresponding Mordell-Weil group over the function
field $\mathbb{Q}(t)$ has rank 1 or 2, respectively. In order to prove this, we
compute the characteristic polynomials of the Frobenius automorphisms acting on
the second $\ell$-adic cohomology groups attached to elliptic surfaces of
Kodaira dimensions 0 and 1.
