ASKSAGE: Sage Q&A Forum - Individual question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Tue, 27 Sep 2016 07:10:35 -0500How to construc a generator point of an elliptic curve?http://ask.sagemath.org/question/9086/how-to-construc-a-generator-point-of-an-elliptic-curve/I have taht elliptic curve defined in sage:
E1=EllipticCurve(GF(8209),[1,0,0,333,6166])
How can i construct a generator point of an elliptic curve?
Thank you very much.Sat, 16 Jun 2012 18:43:54 -0500http://ask.sagemath.org/question/9086/how-to-construc-a-generator-point-of-an-elliptic-curve/Answer by achrzesz for <p>I have taht elliptic curve defined in sage:</p>
<p>E1=EllipticCurve(GF(8209),[1,0,0,333,6166])</p>
<p>How can i construct a generator point of an elliptic curve?</p>
<p>Thank you very much.</p>
http://ask.sagemath.org/question/9086/how-to-construc-a-generator-point-of-an-elliptic-curve/?answer=13720#post-id-13720
sage: E1=EllipticCurve(GF(8209),[1,0,0,333,6166])
sage: P=E1.gen(0);P
(3714 : 7019 : 1)
sage: P.order()
8210
Sun, 17 Jun 2012 07:43:46 -0500http://ask.sagemath.org/question/9086/how-to-construc-a-generator-point-of-an-elliptic-curve/?answer=13720#post-id-13720Comment by vescapam for <pre><code>sage: E1=EllipticCurve(GF(8209),[1,0,0,333,6166])
sage: P=E1.gen(0);P
(3714 : 7019 : 1)
sage: P.order()
8210
</code></pre>
http://ask.sagemath.org/question/9086/how-to-construc-a-generator-point-of-an-elliptic-curve/?comment=19579#post-id-19579Thank you!!!!, i know another method but i think that hhis is better.Sun, 17 Jun 2012 12:29:35 -0500http://ask.sagemath.org/question/9086/how-to-construc-a-generator-point-of-an-elliptic-curve/?comment=19579#post-id-19579Answer by John Cremona for <p>I have taht elliptic curve defined in sage:</p>
<p>E1=EllipticCurve(GF(8209),[1,0,0,333,6166])</p>
<p>How can i construct a generator point of an elliptic curve?</p>
<p>Thank you very much.</p>
http://ask.sagemath.org/question/9086/how-to-construc-a-generator-point-of-an-elliptic-curve/?answer=34970#post-id-34970The group of points on an elliptic curve over a finite field is very often cyclic but may be a product of two cyclic factors, in which case the question is to give (two) generating points rather than (one) generating point. In your example,
sage: E1.abelian_group()
Additive abelian group isomorphic to Z/8210 embedded in Abelian group of points on Elliptic Curve defined by y^2 + x*y = x^3 + 333*x + 6166 over Finite Field of size 8209
shows that the group is cyclic, In any case
sage: E1.gens()
will give the list of at most two generators. For example,
sage: E2=EllipticCurve(GF(8209),[1,0,0,333,0])
sage: E2.gens()
[(7400 : 284 : 1), (4824 : 5797 : 1)]
sage: [P.order() for P in E2.gens()]
[4062, 2]
Tue, 27 Sep 2016 07:10:35 -0500http://ask.sagemath.org/question/9086/how-to-construc-a-generator-point-of-an-elliptic-curve/?answer=34970#post-id-34970