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$\newcommand\Q{\mathbb Q}$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.

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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 $E(\mathbf{R})$:

plot for 43a1

  • 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: plot for 37a1

  • 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.

plot for 359a1

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  • $\begingroup$ +1 Really nice explicit explanation! $\endgroup$ Dec 21, 2021 at 6:16
  • $\begingroup$ I couldn't have asked for a better answer! Thank you David. $\endgroup$
    – quantum
    Dec 21, 2021 at 17:12
  • $\begingroup$ What if we have a curve of rank 2. How would be the points distributed if they were produces just by one independent point (maybe plus torsion points)? Can you also post your code to produce such images? I guess you used Sagemath? $\endgroup$ Oct 2, 2023 at 10:53

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