# Transforming genus zero curve to conic

The following code outputs rational parametrization of genus `0`

curve of degree `5`

:

```
x, y = QQ['x,y'].gens()
C=Curve(2*x^5 + x^2*y - 4*x^3*y + 2*x*y^2 + 2*x*y^3 + 1*y^5)
print(C.is_smooth())
print(C.genus())
C.rational_parameterization()
```

```
False
0
Scheme morphism:
From: Affine Space of dimension 1 over Rational Field
To: Affine Plane Curve over Rational Field defined by 2*x^5 + y^5 - 4*x^3*y + 2*x*y^3 + x^2*y + 2*x*y^2
Defn: Defined on coordinates by sending (t) to
((-2*t^2 - t)/(4*t^5 + 1), (4*t^4 + 2*t^3)/(4*t^5 + 1))
```

Is there a function that instead of parametrization outputs transformation `x -> f1(u, v), y -> f2(u, v)`

so that the curve is transformed into a conic section curve (any curve of degree `2`

)?

The curve would look like this with given coefficients `a1..a6`

.

```
a1*u^2 + a2*v^2 + a3*u*v + a4*u + a5*v + a6
```

**UPDATE:**

Here is an example of a curve of degree `3`

and genus `0`

that I was able to transform to conic - specifically parabola (with forward and backward transformations):

$$y^2=x^3-x^2,\left(x\to \frac{v}{u^2},y\to \frac{v}{u^3}\right)\\ v=u^2+1,\left(u\to \frac{x}{y},v\to \frac{x^3}{y^2}\right)$$

**UPDATE2:**

$$2 x^5-4 x^3 y+x^2 y+2 x y^3+2 x y^2+y^5=0,\left(x\to -\frac{8 u v^3-u+4 v^3-2 v}{16 v^5-1},y\to \frac{2 v \left(8 u v^3-u+4 v^3-2 v\right)}{16 v^5-1}\right)\\v-u^2=0,\left(u\to \frac{-2 x^5+2 x^3 y-x y^3-y^5}{2 x \left(x^3+y^3\right)},v\to -\frac{y}{2 x}\right)$$

How the requested transformation is related to the rational parametrization?

If there does not exist rational parametrization of a curve then it is impossible to transform it birationally to conic curve. So this is the relation. I also think that from the rational parametrization it is possible to compute the transformation to conic, but I have not figured it out yet.

It does not look like just a technical task. You'd better ask a question about a suitable algorithm at https://math.stackexchange.com or alike.

It is likely that the function is already implemented in sage or that combination of two or three functions would produce the result, but it is hard to search the documentation with lots of functions. Wonder if there is some AI that would give reasonable answer to this question.

I see that you got a detailed answer at https://math.stackexchange.com/q/4784138 Now you can ask someone here to translate given Magma code to Sage.