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.Mon, 09 Jan 2017 10:52:15 -0600Elliptic Curve over finite fields - Irreducibility proofhttp://ask.sagemath.org/question/36115/elliptic-curve-over-finite-fields-irreducibility-proof/ Hello,
I am new to Sage and I want to prove an elliptic curve is irreducible.
How can I prove that?
Example:
- E = EllipticCurve(GF(256),[8,4,3,2,0])
Thank youWed, 28 Dec 2016 14:30:26 -0600http://ask.sagemath.org/question/36115/elliptic-curve-over-finite-fields-irreducibility-proof/Answer by nbruin for <p>Hello,
I am new to Sage and I want to prove an elliptic curve is irreducible.
How can I prove that?</p>
<p>Example:
- E = EllipticCurve(GF(256),[8,4,3,2,0])</p>
<p>Thank you</p>
http://ask.sagemath.org/question/36115/elliptic-curve-over-finite-fields-irreducibility-proof/?answer=36234#post-id-36234The word irreducible gets used **a lot** in mathematics. For elliptic curves over finite fields the implementation is just "irreducible as a scheme". Since elliptic curves are irreducible by definition the implementation could just be
def is_irreducible(self):
return True
There are no steps involved.
For elliptic curves *over QQ* `is_irreducible(p)` is still a legacy test of whether the galois representation on the p-torsion is irreducible (but read the documentation about where this is moved). Once this deprecation has been resolved, the removal of the special `is_irreducible` would expose the scheme theoretic routine on elliptic curves over QQ as well.Mon, 09 Jan 2017 10:52:15 -0600http://ask.sagemath.org/question/36115/elliptic-curve-over-finite-fields-irreducibility-proof/?answer=36234#post-id-36234Answer by tmonteil for <p>Hello,
I am new to Sage and I want to prove an elliptic curve is irreducible.
How can I prove that?</p>
<p>Example:
- E = EllipticCurve(GF(256),[8,4,3,2,0])</p>
<p>Thank you</p>
http://ask.sagemath.org/question/36115/elliptic-curve-over-finite-fields-irreducibility-proof/?answer=36124#post-id-36124To know which methods are available to a Python (or Sage) object, you can use tab-completion:
sage: E = EllipticCurve(GF(256),[8,4,3,2,0])
sage: E.<TAB>
Then you will notice that there is a `is_irreducible` method available:
sage: E.is_irreducible()
True
**EDIT** To see how the computation (hence the proof) works, you can get the source code of the method wit two question marks:
sage: E.is_irreducible??
Then you will see that is looks at its defining ideal, and see if it is prime. For this (recursively look at the source code), it looks at its complete primary decomposition, this computation is forwarded to Singular, so you will have to look there to see how that part is done algorithmically.Thu, 29 Dec 2016 13:37:08 -0600http://ask.sagemath.org/question/36115/elliptic-curve-over-finite-fields-irreducibility-proof/?answer=36124#post-id-36124Comment by mrennekamp for <p>To know which methods are available to a Python (or Sage) object, you can use tab-completion:</p>
<pre><code>sage: E = EllipticCurve(GF(256),[8,4,3,2,0])
sage: E.<TAB>
</code></pre>
<p>Then you will notice that there is a <code>is_irreducible</code> method available:</p>
<pre><code>sage: E.is_irreducible()
True
</code></pre>
<p><strong>EDIT</strong> To see how the computation (hence the proof) works, you can get the source code of the method wit two question marks:</p>
<pre><code>sage: E.is_irreducible??
</code></pre>
<p>Then you will see that is looks at its defining ideal, and see if it is prime. For this (recursively look at the source code), it looks at its complete primary decomposition, this computation is forwarded to Singular, so you will have to look there to see how that part is done algorithmically.</p>
http://ask.sagemath.org/question/36115/elliptic-curve-over-finite-fields-irreducibility-proof/?comment=36224#post-id-36224What logical methods are available? Also, *why* do you want to prove this - fun? academic research? etc. ?Sun, 08 Jan 2017 14:28:42 -0600http://ask.sagemath.org/question/36115/elliptic-curve-over-finite-fields-irreducibility-proof/?comment=36224#post-id-36224Comment by emrealparslan for <p>To know which methods are available to a Python (or Sage) object, you can use tab-completion:</p>
<pre><code>sage: E = EllipticCurve(GF(256),[8,4,3,2,0])
sage: E.<TAB>
</code></pre>
<p>Then you will notice that there is a <code>is_irreducible</code> method available:</p>
<pre><code>sage: E.is_irreducible()
True
</code></pre>
<p><strong>EDIT</strong> To see how the computation (hence the proof) works, you can get the source code of the method wit two question marks:</p>
<pre><code>sage: E.is_irreducible??
</code></pre>
<p>Then you will see that is looks at its defining ideal, and see if it is prime. For this (recursively look at the source code), it looks at its complete primary decomposition, this computation is forwarded to Singular, so you will have to look there to see how that part is done algorithmically.</p>
http://ask.sagemath.org/question/36115/elliptic-curve-over-finite-fields-irreducibility-proof/?comment=36197#post-id-36197Thank you for your answer but how do I prove that step by step instead of using is_irreducible() ?Thu, 05 Jan 2017 21:11:37 -0600http://ask.sagemath.org/question/36115/elliptic-curve-over-finite-fields-irreducibility-proof/?comment=36197#post-id-36197