# Get point coordinates of curve over number field

Suppose I have equation of curve `C`

:

```
curve
# => y^2 + (x^2 + x)*y + x
curve.parent()
# => Univariate Polynomial Ring in y over Rational function field in x over Rational Field
```

and I know that there is a point with `x`

value is a root of another equation (i.e. element of corresponding Number Field):

```
equation
# => x^3 + x^2 - 2*x - 9/2
equation.parent()
# => Univariate Polynomial Ring in x over Rational Field
```

is x-coordinate of some point of `C`

.

How to properly get y-coordinate of `C`

in `x`

?

It is not as easy as it seems because of conversion problems.

My workaround is as following:

```
FF.<z> = NumberField(equation)
P.<x,y> = QQ[]
u = P(curve).subs(x=z)
P.<y> = FF[]
return z, P(u).roots()[0][0]
```

and it doesn't seem right.

Are there more elegant way of doing it?

P.S. `curve`

is constructed as follows:

```
F = FunctionField(QQ, 'x')
x = F.gen()
R.<y> = F[]
curve = y^2 + (x^2 + x)*y + x;
```

But it was done in another place so I have no direct access to `x`

and `y`

from above code.

Could you please provide the construction of

`curve`

?@tmonteil, I've added P.S. with details