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The elliptic curve $-4 x^3 + 4 x^2 y + 16 x - y^3 + 9 y$ goes through $21$ integer points in the range $-9$ to $9$. Is that the maximum?

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Is that the maximum in the range $[-9,9]$? Why make that cutoff? – Aaron May 31 '11 at 17:35
[-9,9] is arbitrary. Any non-tangent line going through two rational points on an elliptic curve will go through a third. Multiplying by the GCD gives an elliptic curve going through more integer lattice points, for a sufficiently large lattice. I was wondering about maxima on a small lattice, so I picked [-9,9] arbitrarily. – Silas Pike May 31 '11 at 20:43

This is a bit of a longshot, but have a look at Matthew Baker and Clayton Petsche, Global discrepancy and small points on elliptic curves, and at some of the papers in the references that have titles that suggest they may be relevant to this problem.

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There is a demonstration, Elliptic Curves on a Small Lattice that gives equations for 5 elliptic curves with 21 lattice points on a [-9,9] lattice, and hundreds more with 20-15 points. – Silas Pike Jun 2 '11 at 14:12

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