Given a prime $p$ such that $3$ does not divide $p-1$, what is the order of the elliptic curve over $\mathbb{F}_p$ given by $E\left(\mathbb{F}_p\right)=\left\{ (x,y) \in {\mathbb{F}_p}^2 \;|\; y^2=x^3+7 \right\}$
I thought if I could find the order of a point then I could find multiples of that which lie in the interval given by Hasse's theorem, but it seems like I dont have enough information for this approach. Any hint as to what else I should look at would be appreciated!
The key result: If $p \not \equiv 1 \mod{3}$ then every element of $\mathbb{F}_p$ has exactly one cube root. Prove this, then the problem should be easy.