# Determining order of the group for an elliptic curve defined over a finite field

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

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.

• Note that assignments may not be answered fully on this site, hints go either into the comment section or into an answer. – Maarten Bodewes May 16 at 22:27
• 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. – corpsfini May 16 at 22:31
• @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? – SARTHAK GUPTA May 16 at 22:36
• Hasse is a bound, the exact answer is found via point counting. And this does look like homework. – kodlu May 16 at 23:51