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I try to understand the definition of Identity element of elliptic curve.

For the following elliptic curve over $k = \mathbb{F}_5$:

$$y^{2} = x^{3} + 1$$

The points that in $E$ are shown in the Table of Point Additions:

$$\begin{array}{|c|c|c|c|} \hline +\ & \infty & (0, 1) & (0, 4) & (2, 2)& (2, 3)& (4, 0)\\\hline \infty &\infty & (0, 1) & (0, 4) & (2, 2)& (2, 3)& (4, 0) \\\hline (0, 1) & (0, 1) & (0, 4) &\infty & (2, 3)& (4, 0)& (2, 2) \\\hline (0, 4) & (0, 4)&\infty & (0, 1) & (4, 0)& (2, 2)& (2, 3) \\\hline (2, 2) & (2, 2)& (2, 3) & (4, 0)& (0, 4)&\infty & (0, 1) \\\hline (2, 3) & (2, 3)& (4, 0)& (2, 2) &\infty& (0, 1)& (0, 4) \\\hline (4, 0) & (4, 0)& (2, 2)& (2, 3) & (0, 1)& (0, 4)&\infty \\\hline \end{array}$$

The only point that corresponds to identity element is the point at infinity.

But how can I be sure that's the only point?

Thanks.

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    $\begingroup$ By definition of the group structure on elliptic curves that is the neutral point of the abelian group. $\endgroup$ – DonAntonio Jul 14 at 11:27
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Whenever you have two identity elements they coincide. Let $e$ and $f$ be two identity elements. Then we get $$e = e \cdot f = f,$$ where the first equality holds as $f$ is an identity element and the second one as $e$ is an identity element. Any yes, for elliptic curves the point at infinity is the identity element.

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