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.
 A: You can find some changes of variables to transform a quartic hyperelliptic curve into a Weierstrass equation at 


*

*Page 77 of: Mordell, Diophantine Equations, Academic Press, New York, 1969.

*Page 37 of: L. Washington, Elliptic Curves: Number Theory and Cryptography
(Discrete Mathematics and Its Applications), Chapman & Hall, 2003.
The results are quoted in my article with Scott Arms and Steven Miller, Appendix B, page 17.
A: 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. 
A: You can search for points on this sort of equation (of any degree) using Sage-s interface to Michael Stoll's ratpoints C program, but it is hidden:
sage: from sage.libs.ratpoints import ratpoints
sage: ratpoints([1,0,3,0,1],1000)
[(1, 1, 0), (1, -1, 0), (0, 1, 1), (0, -1, 1)]

Now this is the equation of a curve of genus 1, so if it has any rational points at all then it is isomorphic to its Jacobian which you can put into Weierstrass form using standard formulas.  This can be done in Sage:  try Jacobian?
