# 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 :$y^2=ax^3+bx^2+cx+d$?

-
It is an elliptic curve problem. What transformation does it need? –  Gerry Myerson Mar 19 '13 at 11:56
@Gerry Myerson:I wonder can we turn the equation to the form as $y^2=ax^3+bx^2+cx+d$? –  Hecke Mar 19 '13 at 12:00
Yes, there is a procedure for doing this. Unfortunately, I'm away from my references, and not up to doing it from memory. But...what advantage do you get by turning it into the form you want? –  Gerry Myerson Mar 19 '13 at 12:28
@Gerry Myerson:To get more solutions from a given solution. –  Hecke Mar 19 '13 at 12:34
@Gerry Myerson:I have known how to do it,thanks a lot! –  Hecke Mar 19 '13 at 13:12

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.

-
Thank you,your answer is very clear to understand. –  Hecke Mar 19 '13 at 14:17