1) Function RationalPoints-of-C (C, $C^{\prime }\left(\mathbb{Q}\right)$ ).

2) Initilize found-ponits = empty list.

3) Add $P=\left(a\mathrm{,0}\right)$ to found-points.

4) for each $Q\left(x^{\prime }\mathrm{,}{y}^{\prime }\right)\in C^{\prime }\left(\mathbb{Q}\right)$ such that $x^{\prime }\ne a$ do.

5) Compute $P\left(x\mathrm{,}y\right)\in C\left(\mathbb{Q}\right)$ such that.

6) $x=\frac{1+a{x}^{\prime }}{x^{\prime }}$ and $y=\frac{y^{\prime }}{{x^{\prime }}^4}$ .

7) end for.

8) Add P and $\iota \left(P\right)$ to found-points.

9) return $\mathbb{Q}$ -points of C.