TITLE: Elliptic curves and connected components
QUESTION [14 upvotes]: Are there elliptic curves of positive rank with two real connected components
in which all the rational points lie only on one component?
Concrete examples are really appreciated.

REPLY [19 votes]: Yes. It is not hard to find an example: Take
$$E \colon y^2 = x^3 - 12 x - 1\,.$$
Then $E(\mathbb Q) \cong \mathbb Z$ and $P = (5, 8)$ is a generator (according
to Magma). Since $P$ is on the component of the identity, all rational points are on that component.