The Explicit Upper Bounds of Kolyvagin and Kato

Elliptic Curve defined by y^2 + y = x^3 + x^2 - 12*x + 2 over Rational Field

0.000000000000000

1

False

[]

[(2, '2-torsion'), (3, '3-torsion')]

([2, 7], 49)

[-11, -20, -23, -35]

[48.999572 ... 49.000367]
[48.999633 ... 49.000306]
[48.999572 ... 49.000306]
[48.999633 ... 49.000367]

11a1 ([2, 5], 1) [5]
14a1 ([2, 3], 1) [2, 3]
15a1 ([2], 1) [2, 4]
17a1 ([2], 1) [4]
19a1 ([2, 3], 1) [3]
20a1 ([2, 3], 1) [3, 2]
21a1 ([2], 1) [4, 2]
24a1 ([2], 1) [4, 2]
26a1 ([2, 3], 1) [1, 3]
26b1 ([2, 7], 1) [7, 1]
27a1 (0, 0) [3]
30a1 ([2, 3], 1) [2, 3, 1]
32a1 (0, 0) [4]
33a1 ([2], 1) [2, 2]
34a1 ([2, 3], 1) [6, 1]
35a1 ([2, 3], 1) [1, 3]
36a1 (0, 0) [3, 2]
37a1 ([2], 1) [1]
37b1 ([2, 3], 1) [3]
38a1 ([2, 3], 1) [1, 3]
38b1 ([2, 5], 1) [5, 1]
39a1 ([2], 1) [2, 2]
40a1 ([2], 1) [2, 2]
42a1 ([2], 1) [8, 2, 1]
43a1 ([2], 1) [1]
44a1 ([2, 3], 1) [3, 1]
45a1 ([2], 1) [2, 1]
46a1 ([2], 1) [2, 1]
48a1 ([2], 1) [2, 2]
49a1 (0, 0) [2]
50a1 ([2, 3, 5], 1) [1, 3]
50b1 ([2, 3, 5], 1) [5, 1]
51a1 ([2, 3], 1) [3, 1]
52a1 ([2], 1) [1, 2]
53a1 ([2], 1) [1]
54a1 ([2, 3], 1) [1, 3]
54b1 ([2, 3], 1) [3, 1]
55a1 ([2], 1) [2, 2]
56a1 ([2], 1) [4, 1]
56b1 ([2], 1) [2, 1]
57a1 ([2], 1) [2, 1]
57c1 ([2, 5], 1) [10, 1]
57b1 ([2], 1) [2, 2]
58a1 ([2], 1) [2, 1]
58b1 ([2, 5], 1) [10, 1]
61a1 ([2], 1) [1]
62a1 ([2], 1) [4, 1]
63a1 ([2], 1) [2, 1]
64a1 (0, 0) [4]
65a1 ([2], 1) [1, 1]
66a1 ([2, 3], 1) [2, 3, 1]
66c1 ([2, 5], 25) [10, 5, 1]
66b1 ([2], 1) [4, 1, 1]
67a1 ([2], 1) [1]
69a1 ([2], 1) [2, 1]
70a1 ([2], 1) [4, 2, 1]
72a1 ([2], 1) [2, 4]
73a1 ([2], 1) [2]
75a1 ([2, 5], 1) [1, 1]
75c1 ([2, 5], 1) [5, 1]
75b1 ([2], 1) [1, 2]
76a1 ([2], 1) [1, 1]
77a1 ([2], 1) [2, 1]
77c1 ([2], 1) [1, 2]
77b1 ([2, 3], 1) [6, 1]
78a1 ([2], 1) [2, 1, 1]
79a1 ([2], 1) [1]
80a1 ([2], 1) [2, 2]
80b1 ([2, 3], 1) [1, 2]
82a1 ([2], 1) [2, 1]
83a1 ([2], 1) [1]
84a1 ([2, 3], 9) [3, 3, 2]
84b1 ([2], 1) [1, 1, 2]
85a1 ([2], 1) [2, 1]
88a1 ([2], 1) [4, 1]
89a1 ([2], 1) [1]
89b1 ([2], 1) [2]
90a1 ([2, 3], 1) [2, 2, 3]
90c1 ([2, 3], 1) [4, 4, 1]
90b1 ([2, 3], 1) [6, 2, 1]
91a1 ([2], 1) [1, 1]
91b1 ([2, 3], 1) [1, 1]
92a1 ([2, 3], 1) [3, 1]
92b1 ([2, 3], 9) [3, 1]
94a1 ([2], 1) [2, 1]
96a1 ([2], 1) [2, 2]
96b1 ([2], 1) [2, 2]
98a1 ([2, 3], 1) [2, 2]
99a1 ([2], 1) [2, 1]
99c1 ([2], 1) [2, 1]
99b1 ([2], 1) [4, 1]
99d1 ([2, 5], 1) [1, 1]

11a1 0 [5]
26b1 0 [7]
38b1 0 [5]
50b1 0 [5]
57c1 0 [2, 5]
58b1 0 [2, 5]
66c1 0 [2, 5]
66c1 0 [2, 5]
75c1 0 [5]
110a1 0 [5]
110a1 0 [5]
114c1 0 [2, 5]
118b1 0 [2, 5]
123a1 1 [5]
141a1 1 [7]
155a1 1 [5]
158c1 0 [2, 5]
170c1 0 [3, 7]
174a1 0 [3, 7]
174b1 0 [7]
174b1 0 [7]
182a1 0 [2, 5]
186b1 0 [5]
186b1 0 [5]
190a1 1 [2, 11]
203a1 0 [5]
214a1 1 [7]
238a1 1 [2, 7]
246b1 0 [5]
246b1 0 [5]
258f1 0 [2, 7]
258f1 0 [2, 7]
258c1 1 [2, 5]
262a1 1 [11]
264d1 0 [2, 7]
270b1 0 [3, 5]
274a1 1 [7]
280b1 1 [2, 3, 5]
285a1 1 [2, 5]
286b1 1 [2, 13]
286d1 0 [2, 5]
286d1 0 [2, 5]
302a1 1 [3, 5]
302c1 1 [5]
303a1 1 [2, 7]
309a1 1 [5]
318d1 1 [2, 11]
322d1 1 [2, 5]
325e1 0 [5]
326b1 1 [5]
330d1 0 [2, 7]
345c1 0 [2, 5]
346b1 1 [7]
348d1 1 [3, 7]
350f1 1 [2, 3, 11]
354f1 1 [2, 7]
354e1 0 [2, 11]
357d1 1 [2, 7]
362b1 1 [7]
364a1 1 [3, 5]
366g1 1 [2, 5]
366b1 0 [5]
366b1 0 [5]
366d1 0 [7]
378g1 0 [5]
381a1 1 [5]
395c1 0 [5]
406d1 0 [2, 5]
408d1 1 [2, 5]
414d1 1 [2, 5]
418b1 1 [2, 13]
426a1 0 [5]
426a1 0 [5]
426c1 0 [3, 5]
430b1 1 [5]
430d1 1 [3, 5]
430d1 1 [3, 5]
434d1 1 [2, 5]
442e1 0 [2, 11]
446b1 1 [2, 7]
458b1 1 [2, 5]
462e1 1 [2, 3, 7]
470f1 1 [2, 3, 7]
470c1 1 [2, 7]
474b1 1 [2, 5]
483a1 0 [5]
490g1 1 [2, 5]
490j1 0 [2, 5]
494d1 1 [2, 3, 13]
497a1 1 [5]
498b1 1 [2, 5]
506f1 1 [13]
506d1 1 [5]
522g1 0 [2, 11]
522i1 1 [2, 5]
522j1 1 [2, 13]
522m1 0 [2, 11]
528i1 0 [2, 5]
530c1 1 [2, 5]
537e1 0 [2, 5]
542a1 0 [2, 7]
542b1 1 [7]
546f1 0 [7]
546f1 0 [7]
546f1 0 [7]
546e1 0 [17]
550i1 1 [2, 3, 7]
550j1 1 [2, 11]
550m1 0 [11]
551c1 1 [2, 7]
555b1 0 [3, 5]
558f1 1 [2, 3, 5]
558g1 1 [2, 7]
560b1 0 [5]
560e1 1 [2, 5]
561b1 1 [2, 5]
570j1 0 [2, 7]
570l1 0 [2, 5]
570l1 0 [2, 5]
570l1 0 [2, 5]
570d1 0 [2, 5]
573b1 0 [5]
574g1 1 [11]
574i1 1 [3, 7]
574i1 1 [3, 7]
574j1 0 [5]
574j1 0 [5]
582c1 1 [2, 5]
585i1 1 [2, 7]
588d1 0 [5]
594h1 0 [5]
594d1 1 [2, 5]
595b1 0 [7]
598d1 1 [2, 17]
600e1 1 [2, 3, 7]
605a1 1 [3, 5]
605c1 1 [5]
606f1 0 [5]
606f1 0 [5]
606d1 0 [7]
608e1 1 [2, 5]
615b1 1 [2, 7]
616b1 0 [2, 5]
618f1 1 [7, 11]
618f1 1 [7, 11]
618e1 1 [5]
618d1 1 [2, 3, 5]
620b1 1 [2, 3, 5]
622a1 1 [7]
624i1 0 [2, 5]
624e1 0 [2, 5]
629d1 1 [5]
630g1 0 [2, 7]
642c1 1 [2, 13]
650k1 1 [2, 3, 7]
651a1 0 [2, 5]
658e1 1 [2, 11]
665a1 1 [5]
665d1 1 [2, 5]
665d1 1 [2, 5]
666g1 0 [2, 23]
666e1 1 [2, 13]
666d1 1 [2, 5]
670a1 1 [11]
670c1 1 [5]
670d1 1 [19]
672b1 1 [2, 3, 5]
674c1 1 [31]
678c1 1 [2, 7]
678d1 0 [2, 7]
678d1 0 [2, 7]
681e1 1 [2, 5]
682b1 1 [3, 19]
690g1 0 [2, 7]
690j1 0 [2, 5]
690e1 1 [2, 5]
696c1 1 [2, 5]
700d1 1 [2, 3, 5]
702j1 0 [11]
702k1 1 [3, 7]
702m1 1 [3, 19]
702l1 1 [2, 3, 5]
705b1 1 [3, 5]
705e1 1 [5]
706b1 1 [23]
706d1 1 [2, 5]
710c1 1 [2, 7]
710b1 1 [2, 17]
710d1 0 [2, 5]
710d1 0 [2, 5]
714c1 0 [5]
714e1 0 [7]
715b1 1 [3, 7]
726g1 1 [2, 3, 5]
726e1 1 [2, 5]
730i1 1 [7]
730j1 1 [3, 7]
734a1 0 [2, 5]
735f1 1 [2, 3, 7]
738f1 1 [2, 11]
738h1 0 [2, 7]
738e1 1 [2, 5]
741c1 0 [11]
741d1 0 [2, 5]
742g1 1 [2, 5]
742e1 1 [2, 5]
755f1 0 [13]
760b1 0 [2, 7]
762g1 0 [2, 3, 7]
762g1 0 [2, 3, 7]
762e1 1 [2, 3, 11]
762d1 1 [2, 5]
770c1 0 [2, 5]
774h1 0 [2, 7]
777g1 1 [2, 5]
777e1 1 [5]
780b1 0 [5]
782c1 0 [2, 7]
782e1 0 [2, 5]
786h1 1 [2, 7]
786j1 1 [3, 7]
786m1 0 [3, 5]
786m1 0 [3, 5]
786l1 1 [5, 7]
786l1 1 [5, 7]
794c1 1 [5]
795c1 0 [3, 5]
798g1 1 [2, 3, 5]
798h1 1 [2, 3, 7]
798c1 1 [2, 5]
798d1 1 [2, 5]
804d1 1 [3, 7]
806f1 0 [2, 5]
806f1 0 [2, 5]
806c1 1 [2, 5]
806d1 1 [2, 3, 11]
810g1 0 [3, 5]
814b1 1 [5]
816i1 1 [2, 11]
817b1 1 [2, 5]
822d1 1 [2, 5]
830c1 1 [2, 5]
831a1 1 [2, 5]
834f1 1 [2, 7]
834g1 1 [2, 5]
834g1 1 [2, 5]
834a1 0 [2, 7]
840d1 0 [2, 5]
842b1 1 [13]
850l1 1 [2, 7]
850d1 1 [2, 7]
854d1 1 [2, 3, 7]
858f1 1 [2, 5, 11]
858f1 1 [2, 5, 11]
858k1 0 [2, 7]
858k1 0 [2, 7]
858k1 0 [2, 7]
858l1 0 [2, 7]
861c1 1 [5, 7]
861c1 1 [5, 7]
861b1 1 [17]
861d1 1 [5]
862e1 1 [2, 5]
870f1 1 [2, 5, 7]
870f1 1 [2, 5, 7]
870i1 0 [2, 5]
870i1 0 [2, 5]
870i1 0 [2, 5]
874f1 0 [3, 7]
874e1 1 [5]
874e1 1 [5]
874d1 1 [5]
876b1 1 [3, 5]
880g1 1 [2, 5]
882h1 1 [2, 3, 5]
882j1 0 [2, 5]
885d1 1 [5]
885d1 1 [5]
886e1 1 [5]
886d1 1 [2, 19]
890f1 1 [13]
890g1 1 [5]
890g1 1 [5]
894f1 1 [5]
894g1 1 [7, 11]
894g1 1 [7, 11]
894c1 0 [3, 5]
894e1 1 [2, 23]
897e1 1 [2, 5]
897d1 1 [2, 3, 5]
901e1 1 [3, 5]
905b1 0 [5]
906h1 1 [5, 11]
906h1 1 [5, 11]
906e1 0 [5]
910f1 1 [2, 5, 11]
910f1 1 [2, 5, 11]
910g1 1 [2, 5]
910h1 1 [2, 3, 17]
910k1 1 [2, 5, 7]
910k1 1 [2, 5, 7]
912h1 1 [2, 5]
915a1 0 [7]
918h1 1 [3, 11]
918j1 1 [2, 3, 5]
920a1 1 [2, 3, 5]
924f1 0 [5]
924h1 1 [3, 5]
924b1 1 [3, 5]
924e1 1 [3, 5]
930f1 0 [11]
930h1 1 [2, 3, 5]
930d1 1 [2, 7]
933b1 1 [11]
934b1 0 [3, 5]
938b1 1 [2, 5]
939c1 1 [5]
942c1 1 [2, 5]
946c1 0 [2, 5]
954i1 1 [2, 5]
954h1 1 [2, 7]
954j1 1 [2, 17]
966h1 0 [5]
966b1 0 [5]
974h1 1 [3, 5]
975i1 1 [2, 3, 7]
975j1 1 [2, 5]
978f1 1 [2, 11]
978g1 1 [2, 7]
986e1 1 [2, 5, 7]
986e1 1 [2, 5, 7]
987e1 1 [2, 3, 5]
987d1 0 [7]
988b1 1 [3, 13]
990l1 0 [7]
996b1 1 [3, 13]

(0, 0)

[-47, -95]

[-0.00018561213 ... 0.00018561213]

[2115.9843 ... 2116.0118]