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