# Maximal small lattice points of an elliptic curve

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