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. -- To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to sage-support+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-support URL: http://www.sagemath.org