Context Alice and Bob want to communicate using the Diffie–Hellman-ElGamal system based on the elliptic curve E over $GF_{11}$ given by the equation $y^2 = x^3 + 2x + 5$, with the point $P = (0, 4)$. Bob decided to have $sB = 2$ as his private key. By experimenting in the group $(E, +)$, Alice found out that $5P = (4, 0)$.
(a) Using the above information, show that the order of P in the group $(E, +)$ is equal to $10$. Hint: determine first the order of the point $(4, 0)$ by working out its inverse in $(E, +)$.
There are more questions regarding the actual encryption, but I think I can do those on my own.
My attempt I know the inverse of a point is the same with a change of sign on the y-coordinate. So my guess is $5P = (4, 0) = (4, - 0) = -5P$, then $5P + 5P = 10P =$ (point representing infinity). This means the order is 10? I'm not very confident in this answer.
Any help appreciated!