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-rwxr-xr-xdivisibility_reductions/1k-max3-posrank.sage1671
-rwxr-xr-xdivisibility_reductions/jumps.sage145
-rwxr-xr-xdivisibility_reductions/mod3mod9.sage10
-rwxr-xr-xdivisibility_reductions/test_div_1.sage223
4 files changed, 2049 insertions, 0 deletions
diff --git a/divisibility_reductions/1k-max3-posrank.sage b/divisibility_reductions/1k-max3-posrank.sage
new file mode 100755
index 0000000..9056405
--- /dev/null
+++ b/divisibility_reductions/1k-max3-posrank.sage
@@ -0,0 +1,1671 @@
1
2# Elliptic curves downloaded from the LMFDB downloaded on 30 January 2019.
3# Below is a list called data. Each entry has the form:
4# [a1,a2,a3,a4,a6] (Weierstrass Coefficients)
5
6
7data = [\
8[1,1,0,-82,-305],\
9[0,0,1,-1,0],\
10[0,0,0,1,6],\
11[0,1,0,0,4],\
12[1,1,1,-55,134],\
13[1,1,0,-6,4],\
14[1,0,1,1,0],\
15[1,1,0,1,1],\
16[1,1,0,-7,5],\
17[1,0,0,-1,2],\
18[1,-1,1,-2,0],\
19[0,1,1,-1,-1],\
20[0,-1,1,-1,-2],\
21[0,0,1,-3,0],\
22[0,0,1,-3,2],\
23[0,-1,1,2,-2],\
24[0,1,1,-12,2],\
25[0,1,0,0,1],\
26[0,1,0,2,0],\
27[1,1,0,3,-3],\
28[0,0,0,-1,1],\
29[0,-1,0,16,0],\
30[0,0,0,1,-1],\
31[0,-1,1,-19,39],\
32[0,-1,0,4,-8],\
33[1,1,1,-102,355],\
34[1,1,0,7,-9],\
35[0,1,1,-441,3422],\
36[1,0,0,-7,9],\
37[0,0,1,-21,40],\
38[0,0,0,-412,3316],\
39[0,-1,0,-9,13],\
40[0,1,0,-57,171],\
41[1,-1,1,25,-26],\
42[0,-1,0,3,-11],\
43[0,-1,0,3,-2],\
44[1,-1,0,13,-11],\
45[0,0,0,-7,-2],\
46[1,1,1,-39,-35],\
47[0,-1,1,19,100],\
48[1,-1,1,-7,8],\
49[1,0,1,-15,22],\
50[1,1,0,-33,61],\
51[1,-1,0,-15,-46],\
52[1,-1,0,-19,37],\
53[0,1,0,11,695],\
54[1,0,1,-7,14],\
55[0,1,0,3,11],\
56[1,0,1,-2,1],\
57[1,-1,1,-2,28],\
58[0,1,0,0,-1],\
59[0,0,1,-12,4],\
60[1,-1,0,21,53],\
61[1,-1,0,-935,11229],\
62[1,1,0,-2819,-58803],\
63[1,1,1,-784,8720],\
64[0,0,0,1,1],\
65[0,0,0,-8,16],\
66[1,1,1,-8,9],\
67[1,-1,0,12,-208],\
68[0,-1,0,-383,3012],\
69[0,1,0,6,-43],\
70[1,1,0,81,-27],\
71[0,-1,0,-10,17],\
72[0,1,1,1815,141239],\
73[1,-1,0,-2,1],\
74[1,-1,0,-4,-2],\
75[0,0,0,-6,5],\
76[1,-1,0,-18,36],\
77[0,-1,0,-13,25],\
78[0,-1,0,-12,24],\
79[0,-1,0,-80,304],\
80[0,0,0,-2,4],\
81[1,-1,1,8,-5],\
82[1,1,1,-9,7],\
83[0,-1,0,-544,-4352],\
84[1,-1,0,90,436],\
85[0,0,1,1,-8],\
86[1,1,1,-120,42282],\
87[1,1,1,9,13],\
88[1,0,0,-185,1401],\
89[0,1,1,-4,-2],\
90[1,1,1,-17,-70],\
91[0,0,0,2,1],\
92[1,1,0,-12,12],\
93[1,1,1,-18,415],\
94[0,-1,0,4,-3],\
95[1,-1,1,-374,2949],\
96[1,-1,1,-171,1904],\
97[1,-1,0,-1414,-44027],\
98[1,-1,1,-12,18],\
99[0,1,1,-1006,11952],\
100[0,0,1,-111,450],\
101[1,-1,0,0,2],\
102[0,1,0,-233,1563],\
103[0,0,0,-8,9],\
104[0,0,1,43,-2088],\
105[1,1,0,-1,-2],\
106[0,1,1,9,344],\
107[1,0,1,-1,1],\
108[0,0,0,-27,27],\
109[1,-1,1,2064,18771],\
110[0,1,1,-16649,821406],\
111[0,1,0,-11,11],\
112[0,0,1,-4,-3],\
113[1,0,1,-10758,428760],\
114[1,0,1,1,1],\
115[0,0,1,-12,9],\
116[0,0,1,242,-333],\
117[1,0,1,-722,7396],\
118[1,0,1,706,-64375],\
119[1,0,1,-17,56],\
120[1,0,1,31,20],\
121[1,-1,0,-3,-1],\
122[1,-1,0,-2430,46732],\
123[0,0,0,1468,-2844],\
124[1,-1,1,-11,27],\
125[1,0,1,6,1],\
126[0,1,0,1,2],\
127[0,-1,1,-2,2],\
128[0,-1,0,-4,5],\
129[0,0,0,-67,226],\
130[1,1,0,-59,-201],\
131[1,-1,0,-70,244],\
132[0,1,0,-1,1],\
133[0,0,1,-5,4],\
134[0,1,1,0,2],\
135[1,1,0,-512,4237],\
136[0,-1,1,1,-1],\
137[1,0,0,31,-192],\
138[0,0,1,-75,256],\
139[0,1,0,-1,31],\
140[0,-1,1,-4,-2],\
141[0,1,1,-10,10],\
142[1,0,1,2,0],\
143[0,-1,0,-432,-3316],\
144[1,1,1,2,-1],\
145[0,-1,1,-83,3818],\
146[1,0,1,-3,0],\
147[0,1,0,84,36],\
148[0,1,1,-156,700],\
149[1,1,1,-56,-135],\
150[0,-1,0,-1,2],\
151[0,-1,1,2,0],\
152[0,-1,1,1,0],\
153[1,1,1,13,177],\
154[1,0,0,-19,33],\
155[1,-1,0,-1,1],\
156[1,0,0,-5,4],\
157[0,0,1,-3,4],\
158[0,-1,0,8,-4],\
159[0,1,0,-3,-2],\
160[0,1,0,1,1],\
161[1,0,0,-1,0],\
162[0,0,0,-12,20],\
163[0,0,0,-2,0],\
164[1,0,0,-12,16],\
165[1,1,0,2,1],\
166[1,-1,1,0,0],\
167[0,-1,1,10,6],\
168[1,-1,0,-80,-256],\
169[0,-1,0,-16,29],\
170[1,0,0,-6,9],\
171[1,0,0,1,25],\
172[0,0,0,-584,5444],\
173[1,-1,0,1,-1],\
174[0,-1,1,-5,6],\
175[1,1,0,-794,8289],\
176[1,-1,0,1,1],\
177[1,0,0,-2,1],\
178[1,-1,0,12,35],\
179[1,0,0,-6,4],\
180[0,0,0,-55,157],\
181[1,0,0,-3,2],\
182[0,0,1,-81,290],\
183[1,1,1,-2,0],\
184[0,1,1,-100,406],\
185[1,1,0,-30,52],\
186[0,-1,0,0,1],\
187[1,-1,0,-20,40],\
188[0,1,0,-21,31],\
189[1,-1,0,0,4],\
190[0,0,1,-3,18],\
191[0,-1,0,55,93],\
192[1,1,1,6,7],\
193[0,0,1,-237,1404],\
194[1,0,0,-2,-1],\
195[1,1,0,2,2],\
196[1,1,0,-3,1],\
197[0,-1,0,-45,133],\
198[1,-1,1,-5,20],\
199[1,0,0,-3,-2],\
200[0,0,1,-3,-2],\
201[0,-1,0,4,4],\
202[1,0,1,1,-5],\
203[0,1,1,1,1],\
204[1,0,0,-213,-1208],\
205[1,1,1,-1,0],\
206[1,0,1,-23,39],\
207[0,1,1,-2376,-61851],\
208[0,-1,0,-21,49],\
209[1,-1,0,0,1],\
210[1,-1,0,-454,5812],\
211[1,-1,1,-12,15],\
212[0,-1,0,-8,16],\
213[0,-1,0,0,4],\
214[0,0,1,-9,10],\
215[0,1,0,-50,129],\
216[1,-1,1,-4,-1],\
217[1,0,0,-24,63],\
218[1,0,0,0,-1],\
219[1,-1,0,-5252,-145223],\
220[0,-1,1,-26,68],\
221[1,-1,1,-1487,-12905],\
222[0,0,0,5,10],\
223[1,-1,1,-2,82],\
224[0,0,1,-597,8820],\
225[0,-1,1,-52,-3863],\
226[0,-1,1,-3,2],\
227[1,0,0,-3,0],\
228[1,1,0,-35,-98],\
229[1,-1,0,-5,7],\
230[1,-1,1,-149,749],\
231[1,0,0,-28,-59],\
232[1,-1,1,-117,141],\
233[0,1,1,-8,8],\
234[0,0,0,-48,196],\
235[0,-1,0,2,1],\
236[1,1,1,-16,-15],\
237[1,1,0,-715,7069],\
238[0,-1,0,-1,197],\
239[0,1,0,-16,16],\
240[1,-1,1,-83,595],\
241[0,1,1,-2,-2],\
242[1,0,0,43,-31],\
243[1,1,1,79,335],\
244[1,-1,1,13,-12],\
245[1,-1,0,-29,-635],\
246[1,-1,0,-2846,59156],\
247[1,0,0,-350,2500],\
248[1,-1,0,1,-3],\
249[1,0,1,-80,-275],\
250[1,1,1,4,-1443],\
251[0,0,1,3,-4],\
252[1,0,1,-8,7],\
253[1,1,1,-5,11],\
254[1,0,1,-174,880],\
255[1,1,1,-3,0],\
256[1,-1,1,-61,197],\
257[1,-1,0,16,-10],\
258[0,-1,0,-3,4],\
259[0,1,0,-17,51],\
260[1,-1,1,318,-2367],\
261[1,0,1,-604,-5734],\
262[0,1,1,-11,-16],\
263[1,1,0,28,157],\
264[1,1,0,0,1],\
265[1,1,1,66,-5],\
266[0,0,0,-13,-18],\
267[0,0,1,175,-1344],\
268[1,-1,0,-107,454],\
269[1,-1,1,-2,-26],\
270[1,0,1,-32,-71],\
271[1,1,1,64,258],\
272[0,0,0,-17,27],\
273[1,1,0,-2,-2],\
274[0,-1,0,-1,-1],\
275[1,1,0,-97,281],\
276[0,0,1,-40,48],\
277[1,0,1,-44,-150],\
278[1,0,1,-9,28],\
279[1,0,0,-11,14],\
280[1,-1,0,11,-18],\
281[1,-1,1,1,-2],\
282[1,1,1,-56,1145],\
283[0,0,0,5,-6],\
284[1,-1,0,-153,4909],\
285[0,0,0,-8,4],\
286[0,1,0,8,89],\
287[1,0,1,0,-1],\
288[1,-1,1,22,105],\
289[1,1,1,-70,195],\
290[1,-1,1,-509,4677],\
291[1,-1,0,-9,-19],\
292[0,0,0,-4,4],\
293[1,-1,0,12,-19],\
294[1,-1,1,-9,9],\
295[1,-1,0,-9,-14],\
296[0,-1,0,-5,1],\
297[1,1,1,-12,45],\
298[0,-1,1,-7,10],\
299[0,1,0,-8,8],\
300[1,-1,1,12,87],\
301[1,1,1,1,0],\
302[0,0,0,5,42],\
303[0,-1,0,-4,8],\
304[1,1,0,25,-14],\
305[1,0,1,33924,-387702],\
306[1,0,1,-1,148],\
307[1,0,1,-4,-2],\
308[0,1,1,-269,1628],\
309[1,-1,0,1,5],\
310[0,1,1,-16,-66],\
311[1,1,0,-1693,26434],\
312[0,0,0,4,-4],\
313[0,1,0,-4,0],\
314[1,-1,0,-42,-127],\
315[1,1,0,-108,-432],\
316[1,-1,1,7,-7],\
317[0,0,1,-4,3],\
318[0,0,1,-808,8840],\
319[0,0,0,-192,1028],\
320[1,-1,0,-45,139],\
321[0,1,1,-179,881],\
322[0,1,1,0,0],\
323[0,0,0,-68,-236],\
324[0,-1,0,3,9],\
325[0,0,1,-8,-12],\
326[1,0,0,-267,1521],\
327[1,0,1,11,0],\
328[1,0,1,4,2],\
329[1,-1,1,-15,87],\
330[0,0,0,-56,-4848],\
331[0,0,0,-10,12],\
332[0,0,0,-5,4],\
333[1,1,0,-1,-1],\
334[0,0,0,8,4],\
335[1,1,1,0,-2],\
336[1,1,1,-26,39],\
337[1,0,0,-4,-5],\
338[0,-1,0,2,-7],\
339[0,-1,1,-1,1],\
340[0,1,1,-399,-3184],\
341[0,0,0,-484,-5324],\
342[0,0,1,1,0],\
343[0,1,0,-309,1991],\
344[1,-1,1,-48,147],\
345[1,-1,0,-5,-3],\
346[1,-1,0,18,202],\
347[0,0,0,-3,14],\
348[0,0,1,-2,1],\
349[1,0,1,170,-3237],\
350[0,-1,0,-23,51],\
351[1,0,0,-1,-64],\
352[0,1,0,-6,4],\
353[0,0,1,-10,12],\
354[1,1,0,2,4],\
355[1,-1,1,-28,63],\
356[0,-1,1,-6,8],\
357[0,1,0,160,3188],\
358[1,1,1,-5,0],\
359[0,1,1,7,2],\
360[1,-1,1,-57,222],\
361[1,1,0,-3,-9],\
362[1,0,0,-42,36],\
363[0,-1,1,-1,0],\
364[0,-1,0,-16,32],\
365[1,0,1,-44,106],\
366[0,-1,1,-8,-82],\
367[1,1,1,-118,418],\
368[0,-1,0,-140,753],\
369[1,0,1,-193,1012],\
370[0,1,0,-158,-812],\
371[1,-1,0,-2,2],\
372[0,1,0,-1,3],\
373[1,0,1,-48,-130],\
374[1,1,1,10,11],\
375[0,0,1,2,0],\
376[1,-1,0,-9,-54],\
377[0,-1,1,-5781,175862],\
378[1,1,0,52,-176],\
379[1,0,0,-120,576],\
380[1,1,0,-27,-59],\
381[0,0,0,-13,18],\
382[0,1,0,0,-76],\
383[1,1,1,-21,27],\
384[1,0,0,-36,81],\
385[1,0,0,-1415,20617],\
386[1,1,1,-861,9267],\
387[1,0,0,-247,809],\
388[0,-1,0,-6,9],\
389[0,0,0,32,-212],\
390[1,1,0,-8,6],\
391[0,1,0,-5,-1],\
392[1,1,1,-230,1251],\
393[1,-1,0,-5,6],\
394[0,0,0,4,4],\
395[1,-1,0,44,496],\
396[1,1,1,24,-23],\
397[1,1,0,-13,13],\
398[0,0,0,-3,34],\
399[1,1,0,-75,250],\
400[0,1,0,7,7],\
401[1,-1,1,0,3],\
402[0,1,1,-197,-208],\
403[0,0,1,-2,2],\
404[0,1,1,-42,110],\
405[0,1,1,-2,2],\
406[1,1,1,-7,-3],\
407[0,1,1,3,7],\
408[0,1,0,-2,9],\
409[0,-1,0,-1,5],\
410[1,1,1,-11,9],\
411[1,0,0,-1,9],\
412[0,-1,0,-3,3],\
413[0,-1,0,-33,85],\
414[0,0,0,-7,7],\
415[0,0,1,-2,-1],\
416[1,1,0,-22,-44],\
417[0,-1,1,-5,-16],\
418[0,0,1,6,13],\
419[0,1,1,-6,2],\
420[0,0,1,49,-86],\
421[1,1,1,-1001,12375],\
422[1,1,1,-21,-5],\
423[1,1,0,-2,-12],\
424[1,0,0,4,-3],\
425[0,1,1,-1,1],\
426[0,0,1,-38,90],\
427[1,-1,1,1,7],\
428[1,1,1,1,2],\
429[0,1,1,-310,3364],\
430[1,-1,1,-14,-16],\
431[1,-1,1,-2,2],\
432[0,-1,0,-221,-1191],\
433[1,1,1,1,5],\
434[1,1,0,-4,2],\
435[1,1,1,-539,4592],\
436[0,1,0,-5,7],\
437[1,1,1,-32,65],\
438[1,-1,1,-118,2693],\
439[1,-1,1,1,0],\
440[0,1,1,2,0],\
441[1,-1,1,-5,6],\
442[1,1,1,2,0],\
443[0,0,0,-48,-124],\
444[1,-1,0,-495,-4118],\
445[1,0,1,32,-210],\
446[0,-1,0,-45,25],\
447[1,-1,0,-2,4],\
448[1,0,1,-103,-406],\
449[0,1,0,8,4],\
450[1,-1,0,3,-10],\
451[1,1,1,-4,-4],\
452[1,-1,1,13,1235],\
453[1,-1,0,9,4],\
454[1,-1,0,3,0],\
455[0,-1,1,-10,16],\
456[0,1,0,-1621,24623],\
457[1,-1,1,-4,7],\
458[0,1,1,10,44],\
459[1,-1,0,-24990,1526804],\
460[0,0,0,-3,322],\
461[1,1,1,-2,15],\
462[1,1,1,-14,75],\
463[1,-1,1,4,-34],\
464[0,0,0,24,16],\
465[1,1,1,-4,-3],\
466[1,1,1,126,1167],\
467[0,1,1,4,14],\
468[0,-1,0,11,-47],\
469[0,1,0,-15,25],\
470[1,0,1,-57,-164],\
471[0,0,1,-3,22],\
472[0,-1,0,7,-3],\
473[1,-1,1,0,2],\
474[1,0,1,0,10],\
475[0,1,0,3,-9],\
476[1,-1,0,66,116],\
477[0,-1,0,56,-1415],\
478[0,1,1,-1,-4],\
479[0,1,0,-61,-205],\
480[1,-1,0,-67,216],\
481[1,-1,1,-41,105],\
482[1,-1,0,135,-243],\
483[1,0,1,-5,2],\
484[0,-1,0,-61,205],\
485[0,-1,0,-20,40],\
486[1,-1,1,-248,1563],\
487[0,-1,0,-1373,-19191],\
488[1,-1,0,-54,-243],\
489[0,0,0,-187,991],\
490[0,0,0,-5,2],\
491[1,0,1,6,-20],\
492[1,-1,0,-1773,63909],\
493[1,-1,0,8,291],\
494[1,-1,1,51,117],\
495[0,1,0,-400,-3308],\
496[0,1,0,-276,1676],\
497[1,1,1,-2460,45949],\
498[0,1,1,-33,94],\
499[1,-1,0,-117,166],\
500[1,0,0,-65,201],\
501[0,-1,0,-36,232],\
502[1,1,1,-5,3],\
503[0,0,0,1,-6],\
504[1,0,1,-1,4],\
505[0,1,1,79,-214],\
506[0,1,0,-4,-8],\
507[0,1,0,0,-16],\
508[1,0,1,-22,40],\
509[1,1,0,1,-1],\
510[0,1,0,0,16],\
511[1,-1,1,-55,72],\
512[0,1,1,-18,24],\
513[0,0,1,-1507,4209],\
514[0,0,0,-8,-16],\
515[0,-1,0,-28,68],\
516[1,0,1,-539,4765],\
517[1,-1,1,-1,17],\
518[1,-1,1,-5,2],\
519[0,1,0,-1,51],\
520[0,-1,1,23004,2393001],\
521[0,0,0,800,26500],\
522[0,0,0,5,-2],\
523[1,-1,1,-26,57],\
524[1,-1,0,0,-3],\
525[1,-1,1,-514,4609],\
526[0,1,1,-28,48],\
527[0,1,0,-9,-13],\
528[0,-1,0,-1,17],\
529[1,1,0,-2,1],\
530[1,1,1,4,-3],\
531[1,1,0,13,13],\
532[1,1,1,-42,87],\
533[1,0,0,-86,292],\
534[0,-1,0,-5,-19],\
535[1,-1,0,-771,-8875],\
536[1,-1,0,3,-2],\
537[1,-1,0,-14,24],\
538[0,0,1,18,-7],\
539[0,1,1,-4758,128144],\
540[1,1,0,5,4],\
541[1,1,1,-101,299],\
542[1,-1,0,0,-1],\
543[1,-1,0,-990,-11745],\
544[0,0,1,-2,0],\
545[0,0,0,-5,5],\
546[0,1,1,-116,444],\
547[1,-1,1,-11,10],\
548[0,-1,0,-88,349],\
549[0,1,0,-440,3412],\
550[1,1,1,-10,8],\
551[1,0,0,-1,4],\
552[1,-1,1,-12,36],\
553[1,0,0,12,117],\
554[1,-1,1,-13,21],\
555[0,-1,0,-100,424],\
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1610[1,0,0,-93104,-10942305],\
1611[1,-1,0,-73125,7629336],\
1612[1,0,0,-6616,206471],\
1613[0,-1,0,-443904,113984640],\
1614[0,-1,0,-80008,-8683988],\
1615[0,-1,0,-10480,-409460],\
1616[1,-1,0,-151200,22667386],\
1617[0,0,0,-19443,-1042958],\
1618[0,-1,0,-12801,-553215],\
1619[1,1,1,-20174,-1111138],\
1620[1,0,0,-3009,-61770],\
1621[1,-1,0,-403083,-97454421],\
1622[0,1,0,-12801,553215],\
1623[0,-1,0,-12544,544960],\
1624[1,-1,0,-34965,2525175],\
1625[1,1,1,-42469,-2756140],\
1626[0,0,0,564,-37744],\
1627[1,-1,0,-15003,-1979636],\
1628[1,1,1,6266,-609505],\
1629[1,0,0,-5654,-181467],\
1630[0,-1,0,-26304,1980288],\
1631[1,0,0,1714,14685],\
1632[1,-1,0,5445,533250],\
1633[0,-1,0,1120,-32340],\
1634[0,-1,0,-2008,-295988],\
1635[1,-1,0,4455,201771],\
1636[0,-1,0,-321,-18879],\
1637[0,0,0,5037,-73262],\
1638[1,1,0,15125,-2468750],\
1639[1,1,1,3876,-89910],\
1640[1,-1,0,170217,10295991],\
1641[1,0,0,1341,18228],\
1642[0,1,0,-321,18879],\
1643[0,-1,0,-544,13888],\
1644[1,-1,0,-8820,404950],\
1645[1,1,1,12481,2376092],\
1646[1,1,0,-3250000,-2256492875],\
1647[1,-1,0,-1170000,487402461],\
1648[1,0,0,-105841,13244636],\
1649[0,0,0,-311043,-66769598],\
1650[1,1,0,-198250,-37090625],\
1651[1,0,0,-5391,285606],\
1652[1,-1,0,-71370,8011575],\
1653[0,0,0,-15843,-1441118],\
1654[0,0,1,-13,18],\
1655[1,0,1,-3,2],\
1656[0,-1,1,-2,0],\
1657[0,0,0,-4,1],\
1658[1,0,0,-4,3],\
1659[1,0,0,0,1],\
1660[1,1,1,-15,16],\
1661[1,0,1,-5,0],\
1662[0,-1,1,0,2],\
1663[0,0,0,-7,10],\
1664[0,1,1,-4,2],\
1665[0,1,1,-2,0],\
1666[1,-1,0,-4,4],\
1667[0,1,1,-12,12],\
1668[0,1,1,1,6],\
1669[0,0,0,-19,34],\
1670[0,-1,1,-24,54],\
1671[0,-1,1,-5,-3]]
diff --git a/divisibility_reductions/jumps.sage b/divisibility_reductions/jumps.sage
new file mode 100755
index 0000000..7517464
--- /dev/null
+++ b/divisibility_reductions/jumps.sage
@@ -0,0 +1,145 @@
1from sage.schemes.elliptic_curves.ell_generic import is_EllipticCurve
2
3# Pre-tests on E and P
4def suitable( E, P, ell ):
5 if not is_EllipticCurve(E):
6 print "E is not an elliptic curve"
7 return False
8 if E.base_field() != QQ:
9 print "E is not defined over Q"
10 return False
11 if not P in E:
12 print "P is not in E"
13 return False
14 if P.has_finite_order():
15 print "P has finite order"
16 return False
17 if not is_prime(ell):
18 print "ell is not prime"
19 return False
20 return True
21
22
23# Stupid auxiliary function. Returns higest power of n dividing m.
24def val( n, m ):
25 ret = 0
26 while m % (n^(ret+1)) == 0:
27 ret += 1
28 return ret
29
30def test_jump_kl2_label( label, ell ):
31 E = EllipticCurve( label )
32 if E.rank() < 1:
33 print "E has rank 0"
34 return
35 P = E.gens()[0]
36 test_jump_kl2( E, P, ell )
37
38def test_jump_kl2( E, P, ell ):
39 if not suitable( E, P, ell ):
40 return
41
42 flag = True
43 for p in Primes():
44 if p > 10^3:
45 break
46 if p == ell:
47 # print "Skipping", p, "because = ell"
48 continue
49 if E.discriminant() % p == 0:
50 # print "Skipping", p, "because E has bad reduction"
51 continue
52 if P.reduction(p).order() % ell != 0:
53 # print "Skipping", p, "because red of P is infinitely ell-divisible"
54 continue
55
56 # print "Working with prime p =", p
57
58 F = FiniteField(p)
59
60 for i in range(3):
61 # print "Torsion level", i, "..."
62
63 # We can build the division fields for increasing powers of l
64 # incrementally. To get a division field, we first compute the
65 # splitting field of the division polynomial. We may be off by
66 # a degree 2 extension. If so, by finite field magic we know
67 # exactly which degree 2 extension we need: the unique one!
68 R.<x> = PolynomialRing(F)
69 F.<a> = E.reduction(p).division_polynomial(ell^i).splitting_field()
70 E_red = E.reduction(p).base_extend(F)
71 # extend if necessary
72 k = val(ell,E_red.gens()[0].order())
73 h = val(ell,E_red.order()) - k
74 if k < i or h < i:
75 F.<a> = F.extension(2)
76 E_red = E_red.base_extend(F)
77
78 P_red = E_red.point(P.reduction(p))
79
80 flag = False
81 if P_red.is_divisible_by(ell):
82 # print "P ell-divisible in this torsion level"
83 flag = True
84 break
85
86 if not flag:
87 print "Point not divisible mod", p
88 print "Stopping here"
89 break
90 if flag:
91 print "*********************************"
92 print "*** Candidate counterexample! ***"
93 print "*********************************"
94
95# Wrapper
96def test_jump_den_label( label, ell ):
97 E = EllipticCurve(label)
98 if E.rank() < 1:
99 print "E has rank 0"
100 return
101 P = E.gens()[0]
102 test_jump_den( E, P, ell )
103
104def test_jump_den( E, P, ell ):
105 if not suitable( E, P, ell ):
106 return
107
108 n_primes = 0
109 inf_divisible = 0
110 tot_l_part = 0
111 tot_non_l_part = 0
112
113 for p in Primes():
114 if p > 5*10^3:
115 break
116 if p == ell or E.discriminant() % p == 0:
117 continue
118
119 n_primes += 1
120
121 E_red = E.reduction(p)
122 N = E_red.order()
123 k = val(ell,E_red.gens()[0].order())
124 h = val(ell,N) - k
125 temp_non_l_part = (N / (ell^(h+k)))-1
126 tot_non_l_part += temp_non_l_part
127 temp_l_part = N - temp_non_l_part
128 tot_l_part += temp_l_part
129
130 if P.reduction(p).order() % ell != 0:
131 inf_divisible += 1
132
133 found = RDF( inf_divisible / n_primes )
134 expected = RDF( tot_non_l_part / (tot_l_part+tot_non_l_part) )
135 print "Found: %0.3f, expected: %0.3f"%(found, expected)
136 if abs(found-expected) > 0.05:
137 print "Unexpected density! Checking divisibility in reductions..."
138 test_jump_kl2( E, P, ell )
139
140def test_from_file( filename ):
141 attach(filename)
142 for coord in data:
143 label = EllipticCurve(coord).label()
144 print "Trying curve", label, "with ell =", 3
145 test_jump_den_label(label,3)
diff --git a/divisibility_reductions/mod3mod9.sage b/divisibility_reductions/mod3mod9.sage
new file mode 100755
index 0000000..3009a58
--- /dev/null
+++ b/divisibility_reductions/mod3mod9.sage
@@ -0,0 +1,10 @@
1# Curves with surjective mod3 represetation but not surjective mod 9
2
3data = [\
4 [ 0, 0, 0, -27, -42 ],
5 [ 0, 0, 0, -162, 792 ],
6 [ 0, 0, 1, -135, -604 ],
7 [ 0, 0, 0, -5427, 153882 ],
8 [ 0, 0, 0, -201042, 34695912 ],
9 [ 0, 0, 0, -1126035, 459913278 ],
10 [ 0, 0, 1, -1127379978, -14569799990728 ] ]
diff --git a/divisibility_reductions/test_div_1.sage b/divisibility_reductions/test_div_1.sage
new file mode 100755
index 0000000..c422891
--- /dev/null
+++ b/divisibility_reductions/test_div_1.sage
@@ -0,0 +1,223 @@
1from sage.schemes.elliptic_curves.ell_generic import is_EllipticCurve
2
3# Pre-tests on E and P
4def suitable( E, P, ell ):
5 if not is_EllipticCurve(E):
6 print "E is not an elliptic curve"
7 return False
8 if E.base_field() != QQ:
9 print "E is not defined over Q"
10 return False
11 if not P in E:
12 print "P is not in E"
13 return False
14 if P.has_finite_order():
15 print "P has finite order"
16 return False
17 if not is_prime(ell):
18 print "ell is not prime"
19 return False
20 return True
21
22
23# Stupid auxiliary function. Returns higest power of n dividing m.
24def val( n, m ):
25 ret = 0
26 while m % (n^(ret+1)) == 0:
27 ret += 1
28 return ret
29
30# Valuation of l-divisibility of the point P
31# The parameters h and k describe the ell-part of E(F) (k>=h)
32def divisibility( P, ell, k, h ):
33 v = 0
34 while v != k-1 and P.is_divisible_by(ell^(v+1)):
35 v += 1
36 return v
37
38# label is the Cremona label of an elliptic curve over Q of rank >= 1.
39# ell is a rational prime.
40def test_label( label, ell ):
41 E = EllipticCurve(label)
42 if E.rank() < 1:
43 print "E has rank 0"
44 return
45 P = E.gens()[0]
46 test( E, P, ell )
47
48# E is an elliptic curve over Q and P a point of infinite order on E.
49# ell is a rational prime.
50def test( E, P, ell ):
51 if not suitable( E, P, ell ):
52 return
53
54 # Setting up for small primes cases
55 print "Computations for small primes starting..."
56 small_primes = []
57 vl = []
58 lpart = []
59 for p in Primes():
60 if p > 100:
61 break
62 if p == ell:
63 print "Skipping", p, "because = ell"
64 continue
65 if E.discriminant() % p == 0:
66 print "Skipping", p, "because E has bad reduction"
67 continue
68 if P.reduction(p).order() % ell != 0:
69 print "Skipping", p, "because red of P is infinitely ell-divisible"
70 continue
71 small_primes.append(p)
72 print "Working with prime p =", p
73
74 # "torsion level"
75 F = FiniteField(p)
76 vlj = []
77 lpj = []
78 for i in range(3): ### I WOULD LIKE TO CHANGE THIS TO SOMETHING BIGGER
79 print "Torsion level", i, "..."
80
81 # We can build the division fields for increasing powers of l
82 # incrementally. To get a division field, we first compute the
83 # splitting field of the division polynomial. We may be off by
84 # a degree 2 extension. If so, by finite field magic we know
85 # exactly which degree 2 extension we need: the unique one!
86 R.<x> = PolynomialRing(F)
87 F.<a> = E.reduction(p).division_polynomial(ell^i).splitting_field()
88 E_red = E.reduction(p).base_extend(F)
89 # extend if necessary
90 k = val(ell,E_red.gens()[0].order())
91 h = val(ell,E_red.order()) - k
92 if k < i or h < i:
93 F.<a> = F.extension(2)
94 E_red = E_red.base_extend(F)
95
96 P_red = E_red.point(P.reduction(p))
97
98 print "[Torsion field computed]"
99
100 # l-part of the abelian group E(F_i)
101 #gg = E_red.abelian_group().gens()
102 #if len( gg ) == 1:
103 # lpj.append( ( val(ell,gg[0].order()), 0 ) )
104 #else:
105 # lpj.append( (val(ell,gg[0].order()), val(ell,gg[1].order())) )
106 k = val(ell,E_red.gens()[0].order())
107 h = val(ell,E_red.order()) - k
108
109 lpj.append( ( k, h ) )
110 vlj.append(divisibility(P_red,ell,k,h))
111
112 vl.append(vlj)
113 lpart.append(lpj)
114
115 # Output
116 print "Done!"
117 for i in range(3):
118 print ""
119 print "******************************************************"
120 print "Torsion Level:", i
121 print ""
122 rows = [small_primes, [vl[j][i] for j in range(len(small_primes))],
123 [lpart[j][i] for j in range(len(small_primes))] ]
124 print table(rows)
125 print ""
126
127# Wrapper for densities(E,P,ell)
128def densities_label( label, ell ):
129 E = EllipticCurve(label)
130 if E.rank() < 1:
131 print "E has rank 0"
132 return
133 P = E.gens()[0]
134 densities( E, P, ell )
135
136def ratio( h, k, d, ell ):
137 x1 = max(k-d,0)
138 y1 = max(h-d,0)
139 x2 = max(k-d-1,0)
140 y2 = max(h-d-1,0)
141 up = ell^(x1+y1)-ell^(x2+y2)
142 down = ell^(h+k)
143 if d == 0:
144 return RDF((up+1)/down)
145 return RDF(up/down)
146
147
148# Computes the densities dens[n] of primes such that P is ell^n-divisible mod p
149def densities( E, P, ell ):
150 if not suitable( E, P, ell ):
151 return
152
153 n_primes = 0
154 divisible = []
155 inf_divisible = 0
156
157 # Array for counting divisibility with specified l-part
158 div_part = [[[0 for i in range(10)] for j in range(10)] for k in range(10)]
159 # total number of elements in the reductions that (do not) form the l-parts
160 tot_l_part = 0
161 tot_non_l_part = 0
162
163 for p in Primes():
164 if p > 10^2:
165 break
166 if p == ell or E.discriminant() % p == 0:
167 continue
168
169 n_primes +=1
170 #print "Working with prime", p
171
172 E_red = E.reduction(p)
173 N = E_red.order()
174 k = val(ell,E_red.gens()[0].order())
175 h = val(ell,N) - k
176 temp_non_l_part = (N / (ell^(h+k)))-1
177 tot_non_l_part += temp_non_l_part
178 temp_l_part = N - temp_non_l_part
179 tot_l_part += temp_l_part
180 # Debug
181 # print "Group structure at", p, ":"
182 # print E_red.abelian_group()
183 # print "Our result:", N, (k,h), temp_l_part, temp_non_l_part
184
185 if P.reduction(p).order() % ell != 0:
186 inf_divisible += 1
187 continue
188
189 n = divisibility( P.reduction(p), ell, \
190 val(ell,E.reduction(p).gens()[0].order()), 0 ) # Wrong parameters but ok
191 while len(divisible) < n+1:
192 divisible.append(0)
193 divisible[n] += 1
194 div_part[k][h][n] += 1
195
196 N = len(divisible)
197 divtotal = [0]*N
198 divtotal[N-1] = divisible[N-1] + inf_divisible
199 for i in range(2,N):
200 divtotal[N-i] = divisible[N-i] + divtotal[N-i+1]
201
202 print "Tested primes:", n_primes
203 for i in range(10):
204 for j in range(10):
205 s = 0
206 for l in range(10):
207 s += div_part[i][j][l]
208 if s != 0:
209 print "l-part (%d,%d):"%(i,j)
210 for l in range(10):
211 if div_part[i][j][l] != 0:
212 print "Exactly %d^%d-divisible: %d, expected %0.3f"%\
213 (ell,l,div_part[i][j][l],ratio(i,j,l,ell))
214
215 for n in range(1,N):
216 print "At least %d^%d-divisble: %0.3f density (%d times)"%(ell,n,\
217 RDF(divtotal[n]/n_primes),divtotal[n])
218 print "Infinitely-divisble: %0.3f density (%d times), expected %0.3f"%\
219 (RDF(inf_divisible/n_primes),inf_divisible,\
220 RDF(tot_non_l_part/(tot_l_part+tot_non_l_part)))
221
222
223

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