Find all answers to $3x^{10}=10x^3\pmod{13}$

Question

Number theory related question.

Give all answers to:

$$3x^{10}\equiv 10x^3 \pmod{13}$$

$0$ is obvious but I can't see a good way to draw out $12$.

I've got this so far:

Rearrange to $3x^{10}-10x^3\equiv 0 \pmod{13}$

Factor out $x^3$, to give:

$x^3(3x^7-10)\equiv 0\pmod{13}$

$0$ works because of $x^3$ term

I'm still looking for:

$(3x^7-10)\equiv 0\pmod{13}$

Is there an easy way to find this?

And if I get an answer to this, am I okay to assume there are no other answers between $0$ and $13$?

Answer 1

$3x^{10}\mod{13}=10x^3\mod{13}$

$-10x^{10}\mod{13}=10x^3\mod{13}$

$10(x^3+x^{10})\mod{13}=0\mod{13}$ This means:$\ $ $-x^3\mod{13}=x^{10}\mod{13}$