Before asking my question, consider elliptic curves over complex numbers.


Let $E\_{\tau}=\mathbb{C}/{\Lambda\_{\tau}}$ be an elliptic curve over $\mathbb{C}$, where $\Lambda\_\tau=\mathbb{Z} \oplus \tau \mathbb{Z}$ with $\text{Im} (\tau) > 0$. 


Let $N>3$ be a prime number. Then, 
the number of cyclic subgroups of order $N$ of $E\_\tau$ is $(N+1)$. They can be written of the form $C\_i=\langle\frac{1+i\tau}{N} \rangle$ for $0 \leq i < N$ or $C\_\infty = \langle \frac{\tau}{N} \rangle$. 


Let $Y\_0(N)$ be the (open) modular curve (over $\mathbb{C}$), which classifies the isomorphism classes of pairs $(E, C)$, where $E$ is an elliptic curve over $\mathbb{C}$ and $C$ is the cyclic subgroup of $E$ of order $N$. Then, $Y\_0(N)(\mathbb{C})$ is isomorphic to $\mathbb{H}/\Gamma\_0(N)$, where $\mathbb{H}$ denotes the complex upper half plane and $\Gamma\_0(N)$ denotes the subgroup of upper triangular matrices modulo $N$ in $\text{SL}\_2(\mathbb{Z})$. The points of $Y\_0(N)$ can be written of the following form: 


either $(E\_{\tau}, C\_0)$ with $\tau \in \mathbb{H}/{\Gamma\_0(N)}$, 


or $(E\_\tau, C)$, where $C=C\_i$ with $i=0, 1, \dots, N-1, \infty$ (and $\tau \in \mathbb{H}/{\text{SL}\_2(\mathbb{Z})}$).


(There is a bijection between the above two.)


There is the Atkin-Lehner involution $w\_N$ on $Y\_0(N)$, which sends $(E, C)$ to $(E/C, E[N]/C)$, where $E[N]$ denotes the group of $N$-torsion points on $E$. (Note that $\tau \in \mathbb{H}/\Gamma\_0(N)$ maps to $\frac{-1}{N\tau}$ by $w\_N$.)


In general, there is no reason to have an isomorphism between $w\_N(E, C)$ to $(E, C')$ for some $C'$ (with same $E$). However, that could happen in characteristic $p$. For instance, if $E$ is the only supersingular elliptic curve over $\overline{\mathbb{F}}\_p$ (up to isomorphism) (e.g. $p=5$), then $E/C$ must be isomorphic to $E$ because $E/C$ is also supersingular.


Now, let $p$ be a prime number congruent to $-1$ modulo $3$, which is different from $N$. Let $\tau=e^{2\pi i/3}$ be a 3rd root of unity, and let $E:=E\_\tau$. Then, the defining equation of $E$ is $y^2 = x^3+1$, which is supersingular over $\mathbb{F}\_p$. So, $[(E, C\_i)]$ can be regarded as a point of $Y\_0(N)(\overline{\mathbb{F}}\_{p})$, where $[(E, C\_i)]$ denotes the isomorphism class of $(E, C\_i)$. Note that $w\_N$ also acts on $Y\_0(N)(\overline{\mathbb{F}}\_{p})$. 


My question is as follows: is there an isomorphism $\varphi$ between $E$ and $E/{C\_i}$ (over $\overline{\mathbb{F}}\_p$)? If so, what is the inverse image of $E[N]/{C\_i}$ by $\varphi$? (It must be a cyclic subgroup of order $N$ of $E$, so it is $C\_j$ for some $j$.) That is, is this true:
$$
w\_N[(E, C\_i)] = [(E, C\_j)]? 
$$ 
If so, what's the relation between $i$ and $j$? 



Atkin-Lehner involutions on modular curves over finite fields
Even for $N=2$ it is not true that $E/C\_i$ is necessarily isomorphic to $E$. In fact, the isogeny graph consisting of supersingular elliptic curves in char $p$ as vertices and $2$-isogenies as edges is connected, so as soon as there are more than one supersingular elliptic curve in char $p$, there will be an $i$ with $E/C\_i$ not isomorphic to $E$.


