Divisibility properties of point counts for elliptic curves over finite fields

Question

Let $E$ be an elliptic curve over the finite field $\mathbb{F}_q$ (of characteristic $p$). My questions are about the following ideal $I_E \subset \mathbb{Z}$:

$$I_E=\langle \#E(\mathbb{F}_q), \#E(\mathbb{F}_{q^2}), \#E(\mathbb{F}_{q^3}), \#E(\mathbb{F}_{q^4}), \#E(\mathbb{F}_{q^5}), \dots \rangle$$

For convenience let $n_E$ be the positive generator of $I_E$.

At LMFDB one can view a lot of examples of the numbers $\#E(\mathbb{F}_{q^d})$, which suggest "rules" obeyed by the $n_E$. After inspecting many examples I noticed the following two "rules" in particular:

1. If $E$ is ordinary then $n_E>1$.

Here is an example with $q=5$ and (apparently) $n_E=3$. Here is a supersingular $E$ with $n_E=1$.

2. If $E$ is supersingular then $p \nmid n_E$.

On the other hand, if $E$ is ordinary then frequently $p \mid n_E$, e.g. in this example.

My questions: Are the above "rules" actually correct? If so, why are they true? Is there any conceptual interpretation of the number $n_E$?

Answer 1

As $E(\Bbb F_q)$ is a subgroup of $E(\Bbb F_{q^r})$ for all $r$, then $|E(\Bbb F_q)|\mid|E(\Bbb F_{q^r})|$ for all $r$, and so $n_E=|E(\Bbb F_q)|$,

If we write $|E(\Bbb F_q)|=q+1-a$, where $a$ is the "trace of Frobenius", one characterisation of supersingularity is that $p\mid a$. Therefore $E$ is supersingular iff $n_E\equiv1\pmod p$.

By Hasse, if an elliptic curve over $\Bbb F_q$ has only one point, then $q=2$. There is such a curve; it is supersingular.