Find lift($E_{p^2}$) of an elliptic curve $E_p$ defined in field $F_p$ where $p$ is a prime

Question

How to find $E_{p^2}$ of an elliptic curve $E_p$ defined over finite field $F_p$ where $p$ is a prime number?

Answer 1

Well, so we have $\,y^2=x^3+2x+3\,$ , so what I had in mind is to check particular cases, say:

$$y=1\Longrightarrow f(x):=x^3+2x+2=0\Longrightarrow x= 3\,\,\,\text{is a root}\,\,\pmod 7$$

Now, as you can see here , since $\,f'(3)\neq 0\,$ (i.e., a simple root), we can define

$$t=-\frac{f(3)}{7}f'(3)^{-1}=-5\cdot 29=-5=2\pmod 7\Longrightarrow s=3+2\cdot 7=17$$

is a root of $\,f(x)\pmod{7^2}\,$ , which means the pair $\,(17,1)\,$ satisfies the ellitpci curve in $\,\Bbb F_{7^2}\,$ ...

Note that the other root modulo $\,7\,$ of this case, which is $\,2\,$ , is a double one, so Hensel's lemma cannot help to lift it modulo $\,49\,$ as $\,f'(2)=0\pmod 7\,$ ...