I have this homework problem: Can there be an elliptic curve, view as a projective curve, with no rational points with at least one 0 as a coordinate?

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    $\begingroup$ What about the point at infinity? $\endgroup$ – Pete L. Clark Sep 9 '13 at 13:03
  • $\begingroup$ What is the definition of an elliptic curve in your context ? Do you have a special form to follow or would a genus 1 curve with a general equation be ok ? $\endgroup$ – minar Sep 9 '13 at 19:44
  • $\begingroup$ If an arbitrary form of genus 1 curve is ok, consider the projective curve with equation: $X^2\,Y+X\,Z^2+Y^2\,Z+X^3+Y^3+Z^3$. Check that it has no rational points with at least one $0$ coordinate. Hint: consider points with coordinates in GF(2). $\endgroup$ – minar Sep 9 '13 at 19:54
  • $\begingroup$ I am sorry! What I meant was "exactly one 0" as coordinate. Thank you. $\endgroup$ – jennifer Sep 9 '13 at 20:49
  • $\begingroup$ My definition of elliptic curve (over the rational) is just smooth genus one curve with at least one rational point. $\endgroup$ – jennifer Sep 9 '13 at 20:54

Sure: take an elliptic curve over $\mathbf Q$ with trivial Mordell-Weil group, for example the curve $11a2$ in Cremona's tables (with equation $y^2+y=x^3−x^2−7820x−263580$), and use a projective transformation (over $\mathbf Q$) to send the point at infinity to $(1:1:1)$.


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