I need to find out the order of the group for an elliptic curve. See the image for the question.

enter image description here

The inequality condition after simplification leads to $-2 \sqrt{p} \leq m \leq2 \sqrt{p}.$ Also, the order of the group will be a factor of $m$ (Lagrange theorem).

I found a research article on https://link.springer.com/content/pdf/10.1134/S2070046611020075.pdf

I could still not figure out the exact answer.

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    $\begingroup$ Note that assignments may not be answered fully on this site, hints go either into the comment section or into an answer. $\endgroup$ – Maarten Bodewes May 16 at 22:27
  • $\begingroup$ You might want to revisit how you got $-2\sqrt p \leq m \leq 2 \sqrt p$ from $(\sqrt p -1)^2 \leq m \leq (\sqrt p +1)^2$ since it's incorrect. Then, the right theorem will give you the answer. $\endgroup$ – corpsfini May 16 at 22:31
  • $\begingroup$ @corpsfini Yes, I made a mistake. It's $-2 \sqrt{p} \leq m-(p+1) \leq2 \sqrt{p}$ and it's theorem of Hasse which will give me the answer. But what precisely is the answer? $\endgroup$ – SARTHAK GUPTA May 16 at 22:36
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    $\begingroup$ Hasse is a bound, the exact answer is found via point counting. And this does look like homework. $\endgroup$ – kodlu May 16 at 23:51

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