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According to Hasse-Weil theorem (see, for example, Silverman, Tate, Rational Points on Elliptic Curves, Theorem 4.1) we have:

If $C$ is a non-singular irreducible curve of genus $g$ defined over a finite field $F_p$, then the number of points on $C$ with coordinates in $F_p$ is equal to $p + 1 - \epsilon$, where the "error term" $\epsilon$ satisfies $|\epsilon| \leq 2g\sqrt{p}$.

For my surprise, I found very easily curves satisfying (in principle) the conditions of the theorem but have no rational points. For example, I have used MAPLE to check that in $F_7$, the curve $$C := {x_{1}}^{3} - 2\,{x_{1}}^{2}\,{x_{3}} + {x_{1}}\,{ x_{2}}^{2} + 3\,{x_{1}}\,{x_{2}}\,{x_{3}} + {x_{1}}\,{x_{3}}^{2} - {x_{2}}^{3} - {x_{2}}\,{x_{3}}^{2} + {x_{3}}^{3}$$ (with $x_1$, $x_2$, $x_3$ the projective coordinates), has no rational points. However $C$ is irreducible, non-singular, and of genus $g=1$, and accordingly it seems to meet the conditions in the thorem above, so it should have at least two points, since $8 - 2\,\sqrt{7} > 2$. It has none, though.

What am I missing?

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  • $\begingroup$ According to my recollection and Wikipedia the formal definition of an elliptic curve is a non-singular projective algebraic curve over K with genus 1 and endowed with a distinguished point defined over K. In other words, for the curve to be called elliptic, it needs to have at least one $K$-rational point. In Weierstrass form the point at infinity is often that distinguished point. I'm afraid I don't remember the proper terminology for curves without $K$-rational points. $\endgroup$ Jan 15, 2020 at 22:23
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    $\begingroup$ @JyrkiLahtonen very late on this one, but in case you were still interested, the terminology for a genus $1$ curve $C$ without a point is a "torsor under $E$" where $E$ is the jacobian of $C$ (equiv. a "principal homogeneous space" for $E$). If you know the order $n$ of $[C] \in WC(E/K)$ then you could call it an $n$-covering. $\endgroup$ Mar 26, 2021 at 22:03

1 Answer 1

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According to Magma, and assuming I typed it correctly, your curve is singular, hence the theorem doesn't apply.

> P<x,y,z>:=ProjectiveSpace(GF(7), 2);
> f:=x^3-2*x^2*z+x*y^2+3*x*y*z+x*z^2-y^3-y*z^2+z^3;
> Curve(P, f);
Curve over GF(7) defined by
x^3 + x*y^2 + 6*y^3 + 5*x^2*z + 3*x*y*z + x*z^2 + 6*y*z^2 + z^3
> RationalPoints(Curve(P, f));
{@ @}
> IsSingular(Curve(P,f));                                                                  
true
> Genus(Curve(P,f));
0
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