Sage: Ticket #10730: simon_two_descent -- reports points as being independent, but they are not
https://trac.sagemath.org/ticket/10730
<p>
Check out this
</p>
<pre class="wiki">sage: F.<a> = NumberField(x^2-x-1)
sage: E = EllipticCurve([1,a+1,a,a,0])
sage: E.simon_two_descent()
(0, 1, [(-1 : -a + 1 : 1), (-a : 0 : 1)])
</pre><p>
According to the docs:
</p>
<pre class="wiki">Computes lower and upper bounds on the rank of the Mordell-Weil group, and a list of independent points.
</pre><p>
It output a lower bound of 0, an upper bound of 1, and gave *two* independent points? Clearly something is wrong. In fact, the points output are all torsion and one is a multiple of the other:
</p>
<pre class="wiki">sage: E.torsion_subgroup()
Torsion Subgroup isomorphic to Z/8 associated to the Elliptic Curve defined by y^2 + x*y + a*y = x^3 + (a+1)*x^2 + a*x over Number Field in a with defining polynomial x^2 - x - 1
sage: Q == 4*P
True
sage: v = E.simon_two_descent()
sage: P,Q =v[2]
sage: Q == 4*P
True
sage: P.order()
8
sage: Q.order()
2
</pre><p>
So instead of claiming the output points are independent, claim nothing about them?
</p>
<p>
This is a duplicate of <a class="closed ticket" href="https://trac.sagemath.org/ticket/5153" title="defect: bug in simon_two_descent for elliptic curves (closed: fixed)">#5153</a>.
</p>
en-usSagehttps://trac.sagemath.org/chrome/site/logo_sagemath_trac.png
https://trac.sagemath.org/ticket/10730
Trac 1.1.6nbruinWed, 02 Feb 2011 18:41:41 GMT
https://trac.sagemath.org/ticket/10730#comment:1
https://trac.sagemath.org/ticket/10730#comment:1
<p>
Since it's doing a 2-descent the code should be able to guarantee that the points returned generate E(k)/2E(k) and should be able to test that the points form a basis of E(k)/2E(k) as an F_2-vector space. That's a kind of independence ...
</p>
TicketnbruinWed, 02 Feb 2011 18:54:14 GMT
https://trac.sagemath.org/ticket/10730#comment:2
https://trac.sagemath.org/ticket/10730#comment:2
<p>
Incidentally, from a 2-(isogeny-)descent on E one can conclude that the rank is 0, so while the upper bound returned does not contradict the documentation, the code is not returning the appropriate bound.
In Magma:
</p>
<blockquote class="citation">
<p>
<a class="missing wiki">RankBound?</a>(<a class="missing wiki">EllipticCurve?</a>([1,a+1,a,a,0]));
</p>
</blockquote>
<p>
0
</p>
<blockquote class="citation">
<p>
<a class="missing wiki">AnalyticRank?</a>(E);
</p>
</blockquote>
<p>
0 0.359929
but you probably already knew that.
</p>
TicketrlmWed, 02 Feb 2011 23:22:51 GMT
https://trac.sagemath.org/ticket/10730#comment:3
https://trac.sagemath.org/ticket/10730#comment:3
<p>
See <a class="closed ticket" href="https://trac.sagemath.org/ticket/5153" title="defect: bug in simon_two_descent for elliptic curves (closed: fixed)">#5153</a>
</p>
TicketjdemeyerSun, 09 Oct 2011 10:42:44 GMTstatus, description, component, milestone changed; reviewer, resolution set
https://trac.sagemath.org/ticket/10730#comment:4
https://trac.sagemath.org/ticket/10730#comment:4
<ul>
<li><strong>status</strong>
changed from <em>new</em> to <em>closed</em>
</li>
<li><strong>description</strong>
modified (<a href="/ticket/10730?action=diff&version=4">diff</a>)
</li>
<li><strong>component</strong>
changed from <em>number fields</em> to <em>elliptic curves</em>
</li>
<li><strong>milestone</strong>
changed from <em>sage-4.7.2</em> to <em>sage-duplicate/invalid/wontfix</em>
</li>
<li><strong>reviewer</strong>
set to <em>Robert Miller</em>
</li>
<li><strong>resolution</strong>
set to <em>duplicate</em>
</li>
</ul>
Ticket