# Height of rational points

Hi, I'm looking at this example of enumeration of rational points from the documentation *Enumeration of rational points on projective schemes*. Here I'm considering the entire projective plane.

```
from sage.schemes.projective.projective_rational_point import enum_projective_number_field
u = QQ['u'].0
K = NumberField(u^3 - 5, 'v')
P.<x,y,z> = ProjectiveSpace(K, 2)
enum_projective_number_field(P, bound=RR(5^(1/3)), prec=2^10)
```

The returned result includes several points like `(v : 1/5*v^2 : 1)`

. If I'm not mistaken, this point is of height 25^(1/3) instead of 5^(1/3): for the infinite place the first coordinate provides a 5^(1/3), and for the place 5 the second coordinate provides another 5^(1/3). Is this a bug?

**Edit.** I reviewed the code, it seems that the problem is in the file `schemes/projective/projective_space.py`

, the method `points_of_bounded_height`

of class `ProjectiveSpace_field`

. When enumerating points with bounded height over a number field, it uses the method `elements_of_bounded_height`

: this gives not just algebraic integers but all field elements with bounded height.

I'm not an expert but

gives 0.536479304144700, which is smaller than 5^(1/3).

Thanks for the comment! First of all,

`global_height`

gives the logarithm height, so one should consider exp(0.53...) which is actually equal to 5^(1/3). But this only computes the maximal height among the three coordinates, instead of the height as a homogeneous coordinate. The former is not a well-defined function, for example for (1:1:1)=(2:2:2) it can give both 0=ln(1) and ln(2).My code for enumeration of points using elimination is here if anyone is interested. There is a hacked-up global height function for homogeneous coordinates.