The rational points on the curve: $y^2=ax^4+bx^2+c$.

Question

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.

Answer 1

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.

Answer 2

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.