arithmetic geometry - Families of genus 2 curves with positive rank jacobians

My guess for some examples is the family of (genus 2) hyperelliptic curves y²=degree 6 poly in x passing through n "randomly chosen" rational points (for n=2,3,4,5, or 6). The family of such curves has dimension 7-n, and I would guess that if a hyperelliptic curve has n "random" rational points on it then the Q rank of its Jacobian is usually at least n-1.

An obvious place to look for explicit examples is
MR1406090 Cassels, J. W. S.; Flynn, E. V. Prolegomena to a middlebrow arithmetic of curves of genus 2. London Mathematical Society Lecture Note Series, 230. Cambridge University Press, Cambridge, 1996. ISBN: 0-521-48370-0