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!
Your idea is correct. It follows immediately from Lagrange's theorem and the Hasse bound. Could the group be twice as big or bigger?