# Dense rational points of an elliptic curve

$$\newcommand\Q{\mathbb Q}$$Could anyone please provide me the following two examples of elliptic curve defined over $$\Q$$ (if they exist)

• It has infinite $$\Q$$-rational points and these points are (in the Euclidean metric of the affine plane) dense on the curve
• It has infinite $$\Q$$-rational points and these points are (in the Euclidean metric of the affine plane) not dense on the curve

I guess proving that the points are infinite is just a proof of having a $$>0$$ rank (which may or may not be obvious). I would also be interested in an argument (or a reference) why one concludes the points are/aren't dense.

There are three cases that can arise. The point is that if $$\Delta(E) < 0$$, then $$E(\mathbf{R})$$ is the circle group $$\mathbf{R} / \mathbf{Z}$$, so any infinite subgroup is dense; while if $$\Delta(E) > 0$$, then $$E(\mathbf{R}) \cong (\mathbf{R} / \mathbf{Z}) \times C_2$$, so an infinite subgroup has to be either dense in the whole group, or contained in the identity component and dense in that.
• For elliptic curve 43a1 in the LMFDB tables, the rank is 1 and the discriminant is -43, so $$E(\mathbf{R})$$ is connected, and $$E(\mathbf{Q})$$ is dense in $$E(\mathbf{R})$$:
• For curve 37a1, there are 2 components and the Mordell-Weil group is $$\mathbf{Z}$$, with the generator lying in the non-identity component. So $$E(\mathbf{R})$$ has two connected components and $$E(\mathbf{Q})$$ is dense in both of them, as in this picture:
• For the curve $$y^2 + x y + y = x^{3} - 23 x + 39$$ (confusingly, labelled 359b1 in LMFDB, but 359a1 in Cremona's tables and in Sage), the Mordell-Weil group is $$\mathbf{Z}$$, the discriminant is $$> 0$$, and the generator is in the identity component; so $$E(\mathbf{Q})$$ is infinite but not dense in $$E(\mathbf{R})$$ -- its closure is exactly the identity component of $$E(\mathbf{R})$$, which has index 2. See diagram below; the red dots are the first few rational points ordered by height -- there are none on the "egg-shaped" component.