So we know many useful theorems that help characterize torsion points on elliptic curves over $\mathbb{Q}$ such as the Nagell–Lutz theorem which provides a useful way to find torsion points on $E/\mathbb{Q}$ and Mazur's theorem which characterizes the torsion subgroup of $E(\mathbb{Q})$.


However, does there exist any elliptic curve over $\mathbb{Q}$ with no torsion points other than the point at infinity? Are there any examples of such curves?


Thanks



Elliptic Curves and Torsion Points
Probably the set of $E/\mathbb{Q}$ with $E(\mathbb{Q})=\{O\}$ has density 1/2 (in an suitable way of ordering curves). A recent (and very deep) result of Manjul Bhargava and Arul Shankar proves that this set has positive density, so there are provably a lot of curves with the property that you request. There's a nice overview of the Bhargava-Shankar result, with some of the history, by Bjorn Poonen. It's freely available at


<http://www-math.mit.edu/~poonen/papers/Exp1049.pdf>


