Let $E$ be an elliptic curve over $\mathbb{Q}$.
Let $\ell$ be a prime number such that the reduced curve $\tilde E_{\ell}$ is non singular. Assume that $\tilde E_{\ell}$ admits a subspace $E'_{\ell}$ and $E''_{\ell}=\tilde E_{\ell}/E'_{\ell}$ such that
- $E'_{\ell}\simeq \mathbb{Z}/\ell\mathbb{Z}$, $E''_{\ell}=\mu_{\ell}$
or
- $E'_{\ell}=\mu_{\ell}$, $E''_{\ell}\simeq \mathbb{Z}/\ell\mathbb{Z}$.
$i$) Why in the first case $E$ must be contains an element of order $\ell$?
$ii_a$) Why in the second case I can consider the curve $E/E'_{\ell}$?
$ii_b$) Someone can please explain me why I have an isogeny of degree $\ell$ between $E/E'_{\ell}$ and $E$?
For $ii_a$)-$ii_b$) I think that the quotient is taken over a subgroup of $E$ isomorphic to $E'_{\ell}$ under the reduction map and if it is finite then the curve $E/E'_{\ell}$ referred to the unique elliptic curve isogenous to E such that the kernel of the isogeny $E/E'_{\ell} \longrightarrow E$ is $E'_{\ell}$. So such isogeny is of degree $\ell$. Is this reasoning right? if it is then all is reduced to show the existence of a finite subgroup of $E$ isomorphic to $E'_{\ell}$ under the reduction map.
Assume now that $2$ is a prime where $E$ has a good reduction of height 1. Then $\tilde E_{2}[2] \simeq \mathbb{Z}/2\mathbb{Z}$, so I have only a non trivial point of order 2. Question:
Is this point rational over $\mathbb{F}_2$?