ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Tue, 10 Dec 2013 01:33:15 +0100Elliptic Curve on FF points are expressed as 3 numbers?https://ask.sagemath.org/question/10818/elliptic-curve-on-ff-points-are-expressed-as-3-numbers/Why are elliptic curve points defined as 3 numbers? As in the following sage snippet:
sage: e = EllipticCurve(GF(13),[0,1])
sage: e.points()
[(0 : 1 : 0), (0 : 1 : 1), (0 : 12 : 1), (2 : 3 : 1), (2 : 10 : 1), (4 : 0 : 1), (5 : 3 : 1), (5 : 10 : 1), (6 : 3 : 1), (6 : 10 : 1), (10 : 0 : 1), (12 : 0 : 1)]
I looked at the official sage documentation [here](http://www.sagemath.org/doc/reference/plane_curves/sage/schemes/elliptic_curves/ell_point.html) and can't seem to find an answer. What does each number in the point represent? X,Y and something else I assume.
I don't have a strong background in abstract algebra/number theory so forgive me if this is something obvious.
Mon, 09 Dec 2013 21:14:52 +0100https://ask.sagemath.org/question/10818/elliptic-curve-on-ff-points-are-expressed-as-3-numbers/Answer by nbruin for <p>Why are elliptic curve points defined as 3 numbers? As in the following sage snippet:</p>
<pre><code>sage: e = EllipticCurve(GF(13),[0,1])
sage: e.points()
[(0 : 1 : 0), (0 : 1 : 1), (0 : 12 : 1), (2 : 3 : 1), (2 : 10 : 1), (4 : 0 : 1), (5 : 3 : 1), (5 : 10 : 1), (6 : 3 : 1), (6 : 10 : 1), (10 : 0 : 1), (12 : 0 : 1)]
</code></pre>
<p>I looked at the official sage documentation <a href="http://www.sagemath.org/doc/reference/plane_curves/sage/schemes/elliptic_curves/ell_point.html">here</a> and can't seem to find an answer. What does each number in the point represent? X,Y and something else I assume.</p>
<p>I don't have a strong background in abstract algebra/number theory so forgive me if this is something obvious.</p>
https://ask.sagemath.org/question/10818/elliptic-curve-on-ff-points-are-expressed-as-3-numbers/?answer=15776#post-id-15776Points are represented using projective coordinates, a common tool to work with curves. Most standard references for elliptic curves will mention them. You can see:
sage: e = EllipticCurve(GF(13),[0,1])
sage: e.defining_polynomial()
-x^3 + y^2*z - z^3
If we set (x,y,z)=(X,Y,1) we get the equation Y^2=X^3+1, which is probably the model you were expecting. An additional solution is (x,y,z)=(0,1,0). That's the "extra point at infinity" that is the identity for the standard group law on an elliptic curve given by an equation of this form.Tue, 10 Dec 2013 01:33:15 +0100https://ask.sagemath.org/question/10818/elliptic-curve-on-ff-points-are-expressed-as-3-numbers/?answer=15776#post-id-15776