Let $E_n$ be the elliptic curve defined by $y^2=x^3-n^2x$, and let $\mathbb{F}_{p^r}$ be a finite field of $p^r$ elements, where $p$ is prime and divides $n$. How can I determine how many points the curve has on such field? I found this problem on the book "Introduction to elliptic curves and modular forms" chapter II: I'm telling this because the book explicitly states that no tool coming from algebraic geometry will be needed so I think it is possible to find a solution to this problem without any algebraic geometry.
My attempt up to now has led me to approach the solution of the case $r=1$ (at least I hope): I considered two cases:
$\bullet$ (case 1) $p\mid 2$ and $p \nmid n$ $\Longrightarrow$ $p=2$ $\Longrightarrow$ $E_n:y^2=x^3-x$
In this case $E_n(\mathbb{F}_2)=\{(0,0),( 1,0)\}$ and so $\#E_n(\mathbb{F}_2)=2$
$\bullet$ (case 2) $p \mid n$ $\Longrightarrow$ $E_n:y^2=x^3$
$\mathbb{F}_p = \{0,1,\ldots,p-1\}$ and the points on the curve are of the type $(a^3,a^2)$ $\forall a \in \mathbb{F}_p$ and are all different since:
if $a^2=b^2$ with $a\neq b$ then because $p$ is prime $a^2a\neq b^2b$ and so $a^3 \neq b^3$. So they are exactly $p$ points.
- Is my attempt correct or at least in the right direction?
- Independently on the answer to the previous question, what's the solution to the whole problem, so including also the cases $r\neq 1$?