ASKSAGE: Sage Q&A Forum - Individual question feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Fri, 14 Jul 2017 12:31:10 -0500Map between projective curves defined in an extension fieldhttps://ask.sagemath.org/question/38263/map-between-projective-curves-defined-in-an-extension-field/For example, suppose I have the following 2 projective curves:
k = GF(13)
x,y,z = ProjectiveSpace(k, 2, 'x,y,z').gens()
E = Curve(2*x^2 + 8*y*z + 8*z^2)
W = Curve(x^2 + y*z + z^2)
I like to define a map from E to W that involves $\sqrt 2$ and $\sqrt 8$, which do not exist in k = GF(13), but do in an extension of k:
x = PolynomialRing(k,'x').gen()
K = GF(13**2, 'w', modulus=x^2-2)
w = K.gen()
So $w = \sqrt 2$ and $2w = \sqrt 8$. The map I like to define sends $(x:y:z)$ to $(wx:2wy:2wz)$.
In this particular example, it's obvious that $(wx:2wy:2wz) = (x:2y:2z)$; but it's just a simple example do demonstrate the problem.
Something like this doesn't work:
x,y,z = ProjectiveSpace(k, 2, 'x,y,z').gens() #or ProjectiveSpace(K, 2, 'x,y,z').gens()
E.Hom(W)([w*x, 2*w*y, 2*w*z])
Thank you.Fri, 14 Jul 2017 11:08:01 -0500https://ask.sagemath.org/question/38263/map-between-projective-curves-defined-in-an-extension-field/Comment by dan_fulea for <p>For example, suppose I have the following 2 projective curves:</p>
<pre><code>k = GF(13)
x,y,z = ProjectiveSpace(k, 2, 'x,y,z').gens()
E = Curve(2*x^2 + 8*y*z + 8*z^2)
W = Curve(x^2 + y*z + z^2)
</code></pre>
<p>I like to define a map from E to W that involves $\sqrt 2$ and $\sqrt 8$, which do not exist in k = GF(13), but do in an extension of k:</p>
<pre><code>x = PolynomialRing(k,'x').gen()
K = GF(13**2, 'w', modulus=x^2-2)
w = K.gen()
</code></pre>
<p>So $w = \sqrt 2$ and $2w = \sqrt 8$. The map I like to define sends $(x:y:z)$ to $(wx:2wy:2wz)$.</p>
<p>In this particular example, it's obvious that $(wx:2wy:2wz) = (x:2y:2z)$; but it's just a simple example do demonstrate the problem.</p>
<p>Something like this doesn't work:</p>
<pre><code>x,y,z = ProjectiveSpace(k, 2, 'x,y,z').gens() #or ProjectiveSpace(K, 2, 'x,y,z').gens()
E.Hom(W)([w*x, 2*w*y, 2*w*z])
</code></pre>
<p>Thank you.</p>
https://ask.sagemath.org/question/38263/map-between-projective-curves-defined-in-an-extension-field/?comment=38265#post-id-38265The following works for me:
sage: K.<a> = GF(13^2)
sage: x,y,z = PolynomialRing(K, names='x,y,z').gens()
sage: E = Curve(2*x^2 + 8*y*z + 8*z^2)
sage: W = Curve(x^2 + y*z + z^2)
sage: E.Hom(W)([a*x, 2*a*y, 2*a*z])
Scheme morphism:
From: Projective Conic Curve over Finite Field in a of size 13^2 defined by 2*x^2 - 5*y*z - 5*z^2
To: Projective Conic Curve over Finite Field in a of size 13^2 defined by x^2 + y*z + z^2
Defn: Defined on coordinates by sending (x : y : z) to
((a)*x : (2*a)*y : (2*a)*z)
sage:
The variables `x,y,z` are in my case generators of a polynomial ring.Fri, 14 Jul 2017 12:31:10 -0500https://ask.sagemath.org/question/38263/map-between-projective-curves-defined-in-an-extension-field/?comment=38265#post-id-38265