Does $y^2 = x^6 - 3x^5 + 3x^4 + 10x^3 + 3x^2 - 3x + 1$ have any rational solutions?

Question

Does $y^2 = x^6 - 3x^5 + 3x^4 + 10x^3 + 3x^2 - 3x + 1$ have any rational solutions? I have some reasonable pre/post graduate Math skills but no access to Magma etc. I suspect there are none other than the obvious 4 (0,1) (0,-1) (-1,2) (-1,-2), but I'd be interested to know for sure (if at all possible). I'm also mainly interested in $x>0$ (if that helps in anyway).

I'm also wondering if the fact that it is palindromic is of any consequence?

Answer 1

One most obvious, is $x=0 \implies y=\pm 1$.

Answer 2

And another obvious solution: if we denote $p(x)$ the r.h.s., we have $p(-1)=4$; whence the solutions $\;(-1,-2)$ and $(-1, 2)$.