Rational Parameterization of Quartic Curve (Variety)

Question

I am trying to find a rational parameterization of the curve (variety) $$x^4+a^2x^2=x^2y^2+(h^2+a^2)y^2$$ If I have done my math correctly, this curve has a singularity of multiplicity 2 at the origin, and the projective extension $$x^4+a^2x^2z^2=x^2y^2+(h^2+a^2)y^2z^2$$ has an isolated singularity at (0:1:0). So, the geometric genus should be $$\frac{(4-1)(4-2)}{2}-2-1=0$$ so the curve should have a rational parameterization. I have found a (non-rational) parameterization by parameterizing the associated equation (hyperbola) found by substituting $$u=x^2$$ $$v=y^2$$ which is: $$(x,y)=(\sqrt{-\frac{a^2-(a^2+h^2))t}{1-t}},\sqrt{-t\frac{a^2-(a^2+h^2)t}{1-t}})$$ but I have not been able to find a substitution that will eliminate the radicals and give me a rational parameterization.

Answer 1

The computation of the genus seems to be incorrect --- the singularity at $(0:0:1)$ only decreases the genus by 1.

In fact, the curve is elliptic. This can be checked by using the Riemann-Hurwitz formula for the map $$ (x:y:z) \mapsto (x^2:y^2:z^2) $$ which is a $4:1$ over the curve $$ X^2 + a^2XZ = XY + (h^2 + a^2)YZ, $$ which is a smooth rational curve (provided $h \ne 0$) and branched over the points $(0:0:1)$, $(0:1:0)$, $(-a^2:0:1)$, and $(1:1:0)$.