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May i ask you for a little help for a problem about elliptic curves? Here's the problem:

Given an elliptic curve $E$ over the finite field $\mathbb{F}_{101}$. We know that there is a point of order $116$ on this curve. Find $\left | E(\mathbb{F}_{101}) \right |$.

OK, i've noticed that $E(\mathbb{F}_{101})$ is a subgroup of $E$. I guess here i should apple the Lagrange's theorem, which says that the order of the subgroup divides the order of the group. How can i use the given order of the point? Do i have to apply the Hasse's theorem?

I would be really glad, if someone could help me with this problem. Thank you in advance! Have a nice day!

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  • $\begingroup$ Yes, you do have to apply Hasse's theorem, but it's really not that arduous here! BTW, $E(F_{101})$ just means $E$ in this context. $\endgroup$ – TonyK Nov 24 '13 at 13:17
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Your idea is correct. It follows immediately from Lagrange's theorem and the Hasse bound. Could the group be twice as big or bigger?

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  • $\begingroup$ I think not, because the order of the multiplicative group of $F_{p}$ is $p-1$? I applied the Hasse bound and got that the number of points lays in the interval $[82,122]$. The only number from this interval, which is divisible by $116$ is $116$ itself. So, the order of $E(F_{101})$ must be 116? $\endgroup$ – Lullaby Nov 24 '13 at 18:37
  • $\begingroup$ @Lullaby: Yes, exactly. $\endgroup$ – TonyK Nov 25 '13 at 8:14

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