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### 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 (the forum does not allow me to include the actual link to the page...). 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?

 2 None slelievre 17634 ●22 ●159 ●348 http://carva.org/samue...

### 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 (the forum does not allow me to include the actual link to the page...). . 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')
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?

### 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.