Consider the elliptic curves $A_0: y^2 + y= x^3-x^2-10x-20 \\$,
$A_1: y^2 + y= x^3-x^2 \\$,
$A_2: y^2 + y= x^3-x^2-7820x-263580 \\$.
We know that $A_0=A_2/\mu_5$, $A_1= A_0/\mu_5$ and $A_2=A_0/(\mathbb{Z}/5\mathbb{Z})$.
How do we conclude the following:
$A_2[5^\infty]$ contains a cyclic subgroup of order 25.
There are non-split exact sequences of $G_{\mathbb{Q}}$-modules:
$0\rightarrow \mathbb{Z}/5\mathbb{Z} \rightarrow A_1[5]\rightarrow \mu_5 \rightarrow 0$
$0\rightarrow \mu_5 \rightarrow A_2[5]\rightarrow \mathbb{Z}/5\mathbb{Z} \rightarrow 0$