Hi list,
I have noticed that D. Simon's algebraic 2-descent does not
work properly for some elliptic curves defined over real quadratic
fields. So I'd like to forward his message to this list.
I think that this defect seems similar to those reported in ticket #9322, #8829,
but I'm not a specialist in these topics, am greatly appreciate your help.
best regards.
---------- Forwarded message ----------
From: <s-yokoy...@math.kyushu-u.ac.jp>
Date: 2011/1/5
To: Iwao Kimura <i...@sci.u-toyama.ac.jp>
Bug-report: Simon's two-descent alg.
this algorithm does return bounds on the rank of
Mordell-Weil group, and a list of independent points.
So I want to compute when K=QQ(sqrt(43)) : real quad.
However, it doesn't work (nor on Sage Notebook).
> Q43.<a>=QuadraticField(43);
> eps = UnitGroup(Q43).fundamental_units()[0];
> E = EllipticCurve(Q43, [0, 1728*eps]);
> dscnt = E.simon_two_descent(verbose=1); dscnt
::
courbe elliptique : Y^2 = x^3 + Mod(917568*y - 6016896, y^2 - 43)
[ omitted ]
*** array index (1) out of allowed range [none]:
*** ...iv,r=nfsqrt(nf,norm(zc))[1];if(DEBUGLEVEL_ell
^--------------------
------------------------------------------------------------------------
---
NameError Traceback (most recent call
last)
/Users/yokoemon/<ipython console> in <module>()
/Applications/sage/local/lib/python2.6/site-packages/sage/schemes/
elliptic_curves/ell_number_field.pyc in simon_two_descent(self, verbose,
lim1, lim3, limtriv, maxprob, limbigprime)
196 t = simon_two_descent(self,
197 verbose=verbose, lim1=lim1, lim3=
lim3, limtriv=limtriv,
--> 198 maxprob=maxprob, limbigprime=
limbigprime)
199 prob_rank = Integer(t[0])
200 two_selmer_rank = Integer(t[1])
/Applications/sage/local/lib/python2.6/site-packages/sage/schemes/
elliptic_curves/gp_simon.pyc in simon_two_descent(E, verbose, lim1, lim3,
limtriv, maxprob, limbigprime)
110 def _gp_mod(*args):
111 return args[0]
--> 112 ans = sage_eval(v, {'Mod': _gp_mod, 'y': K.gen(0)})
113 inv_transform = F.isomorphism_to(E)
114 ans[2] = [inv_transform(F(P)) for P in ans[2]]
/Applications/sage/local/lib/python2.6/site-packages/sage/misc/sage_eval.
pyc in sage_eval(source, locals, cmds, preparse)
197 return locals['_sage_eval_returnval_']
198 else:
--> 199 return eval(source, sage.all.__dict__, locals)
200
201
/Applications/sage/local/lib/python2.6/site-packages/sage/all.pyc in <
module>()
NameError: name 'ans' is not defined
::
if K=QQ(sqrt(41)), it works well.
> Q41.<a>=QuadraticField(41);
> eps = UnitGroup(Q41).fundamental_units()[0];
> E = EllipticCurve(Q41, [0, 1728*eps]);
> dscnt = E.simon_two_descent(verbose=1); dscnt
courbe elliptique : Y^2 = x^3 + Mod(8640*y + 55296, y^2 - 41)
points triviaux sur la courbe = [[1, 1, 0]]
point trouve = [Mod(-377788/93025*y - 1952448/93025, y^2 - 41),
Mod(-205379776/28372625*y - 1032512096/28372625, y^2 - 41)]
#S(E/K)[2] = 4
#E(K)/2E(K) >= 2
#III(E/K)[2] <= 2
rang(E/K) >= 1
III devrait etre un carre, donc
#E(K)/2E(K) = 4
#III(E/K)[2] = 1
rang(E/K) = 2
listpointsmwr = [[Mod(-377788/93025*y - 1952448/93025, y^2 - 41),
Mod(-205379776/28372625*y - 1032512096/28372625, y^2 - 41)]]
(2, 2, [(-377788/93025*a - 1952448/93025 : -205379776/28372625*a -
1032512096/28372625 : 1)])
If someone have any comments,
would you mind telling me about the problem??
Best wishes,
Shun'ichi Yokoyama
Faculty of Mathematics, Kyushu University, Japan
s-yokoy...@math.kyushu-u.ac.jp
--
---
Iwao KIMURA
Dept. Math. University of Toyama, Japan.
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