The rational points on the curve: $y^2=ax^4+bx^2+c$. I wonder how to find the rational points on the curve: $y^2=ax^4+bx^2+c$.
Is there infinite rational points on this curve?
For example:$y^2=x^4+3x^2+1.$If we set $y=x^2+k$,then $2kx^2+k^2=3x^2+1$, Can one turn the equation to the form :$y^2=ax^3+bx^2+cx+d$?
Thanks in advance.

You can turn $y^2 = a x^4 + b x^2 + c$ into $y^2 = x^3 + px + q$ assuming you can find one rational point on $y^2 = a x^4 + b x^2 +c$. The easiest case is when $a$ is square. I do an example of this computation here. 
