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.Thu, 12 Dec 2019 11:29:30 -0600Sagemath refuses to load singular curvehttps://ask.sagemath.org/question/48982/sagemath-refuses-to-load-singular-curve/I know that this below Elliptic curve is singular at $(23796,0)$
F = GF(23981)
E = EllipticCurve(F,[0, 0, 0, 17230, 22699])
Since
$\frac{\partial f}{\partial x} = 3x^2 + 17230 =0 \pmod p$ and vanishes at $x=\{185,23796\}$ and $(185,0)$ not on the curve.
$\frac{\partial f}{\partial y} = -2y = 0 \pmod p$ and vanishes at $y=0$
I'm trying to define this curve to plot but SageMath gives the error;
> ArithmeticError: invariants (0, 0, 0, 17230, 22699) define a singular curve
How can I plot this curve?Mon, 09 Dec 2019 14:20:27 -0600https://ask.sagemath.org/question/48982/sagemath-refuses-to-load-singular-curve/Answer by nbruin for <p>I know that this below Elliptic curve is singular at $(23796,0)$</p>
<pre><code>F = GF(23981)
E = EllipticCurve(F,[0, 0, 0, 17230, 22699])
</code></pre>
<p>Since </p>
<p>$\frac{\partial f}{\partial x} = 3x^2 + 17230 =0 \pmod p$ and vanishes at $x={185,23796}$ and $(185,0)$ not on the curve.</p>
<p>$\frac{\partial f}{\partial y} = -2y = 0 \pmod p$ and vanishes at $y=0$ </p>
<p>I'm trying to define this curve to plot but SageMath gives the error;</p>
<blockquote>
<p>ArithmeticError: invariants (0, 0, 0, 17230, 22699) define a singular curve</p>
</blockquote>
<p>How can I plot this curve?</p>
https://ask.sagemath.org/question/48982/sagemath-refuses-to-load-singular-curve/?answer=48987#post-id-48987You can define the curve easily; just not as an elliptic curve, because being nonsingular is part of their definition.
sage: F = GF(23981)
sage: A.<x,y>=F[]
sage: C=Curve(y^2-(x^3+17230*x+22699))
Plotting a curve over a finite field will be difficult and probably uninsightful.Tue, 10 Dec 2019 09:54:37 -0600https://ask.sagemath.org/question/48982/sagemath-refuses-to-load-singular-curve/?answer=48987#post-id-48987Comment by nbruin for <p>You can define the curve easily; just not as an elliptic curve, because being nonsingular is part of their definition.</p>
<pre><code>sage: F = GF(23981)
sage: A.<x,y>=F[]
sage: C=Curve(y^2-(x^3+17230*x+22699))
</code></pre>
<p>Plotting a curve over a finite field will be difficult and probably uninsightful.</p>
https://ask.sagemath.org/question/48982/sagemath-refuses-to-load-singular-curve/?comment=49017#post-id-49017`F['x'](x^3+17230*x+22699).factor()` gives `(x + 23611) * (x + 185)^2`.
it's a node. The "picture" that you're used to over the reals doesn't apply over other fields (well ... for subfields over the reals it still helps a bit). We only use the same terminology over other fields because we have algebraic characterizations of the different types of singularities.
In this case, using the tangent cone is probably the easiest: move the singularity to the origin:
sage: f=y^2-(x^3+17230*x+22699)
sage: f(x-185,y)
-x^3 + 555*x^2 + y^2
You can see that the lowest degree non-zero homogeneous part is `555*x^2+y^2`. Its degree is larger than 1, so no single tangent line: singular point. The form you do get does not have repeated roots, so the singularity is a node (of degree 2).Thu, 12 Dec 2019 11:29:30 -0600https://ask.sagemath.org/question/48982/sagemath-refuses-to-load-singular-curve/?comment=49017#post-id-49017Comment by kelalaka for <p>You can define the curve easily; just not as an elliptic curve, because being nonsingular is part of their definition.</p>
<pre><code>sage: F = GF(23981)
sage: A.<x,y>=F[]
sage: C=Curve(y^2-(x^3+17230*x+22699))
</code></pre>
<p>Plotting a curve over a finite field will be difficult and probably uninsightful.</p>
https://ask.sagemath.org/question/48982/sagemath-refuses-to-load-singular-curve/?comment=49012#post-id-49012The shape of the singular point. Cusp or node. Of course, when drawing over the finite field, we got only point. Should I look at the curve over Q?Thu, 12 Dec 2019 07:43:12 -0600https://ask.sagemath.org/question/48982/sagemath-refuses-to-load-singular-curve/?comment=49012#post-id-49012Comment by nbruin for <p>You can define the curve easily; just not as an elliptic curve, because being nonsingular is part of their definition.</p>
<pre><code>sage: F = GF(23981)
sage: A.<x,y>=F[]
sage: C=Curve(y^2-(x^3+17230*x+22699))
</code></pre>
<p>Plotting a curve over a finite field will be difficult and probably uninsightful.</p>
https://ask.sagemath.org/question/48982/sagemath-refuses-to-load-singular-curve/?comment=49004#post-id-49004What picture do you expect? What do you think you'll be able to tell from the picture?Wed, 11 Dec 2019 18:40:53 -0600https://ask.sagemath.org/question/48982/sagemath-refuses-to-load-singular-curve/?comment=49004#post-id-49004Comment by kelalaka for <p>You can define the curve easily; just not as an elliptic curve, because being nonsingular is part of their definition.</p>
<pre><code>sage: F = GF(23981)
sage: A.<x,y>=F[]
sage: C=Curve(y^2-(x^3+17230*x+22699))
</code></pre>
<p>Plotting a curve over a finite field will be difficult and probably uninsightful.</p>
https://ask.sagemath.org/question/48982/sagemath-refuses-to-load-singular-curve/?comment=48999#post-id-48999When I try to plot, I've the following error `NotImplementedError: plotting of curves over Finite Field of size 23981 is not implemented yet`Wed, 11 Dec 2019 13:41:15 -0600https://ask.sagemath.org/question/48982/sagemath-refuses-to-load-singular-curve/?comment=48999#post-id-48999