1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
|
diff --git a/src/sage/libs/eclib/interface.py b/src/sage/libs/eclib/interface.py
index e898456720..6b98c12328 100644
--- a/src/sage/libs/eclib/interface.py
+++ b/src/sage/libs/eclib/interface.py
@@ -758,78 +758,78 @@ class mwrank_MordellWeil(SageObject):
sage: EQ = mwrank_MordellWeil(E, verbose=True)
sage: EQ.search(1)
- P1 = [0:1:0] is torsion point, order 1
- P1 = [-3:0:1] is generator number 1
- saturating up to 20...Checking 2-saturation
+ P1 = [0:1:0] is torsion point, order 1
+ P1 = [-3:0:1] is generator number 1
+ saturating up to 20...Checking 2-saturation...
Points have successfully been 2-saturated (max q used = 7)
- Checking 3-saturation
+ Checking 3-saturation...
Points have successfully been 3-saturated (max q used = 7)
- Checking 5-saturation
+ Checking 5-saturation...
Points have successfully been 5-saturated (max q used = 23)
- Checking 7-saturation
+ Checking 7-saturation...
Points have successfully been 7-saturated (max q used = 41)
- Checking 11-saturation
+ Checking 11-saturation...
Points have successfully been 11-saturated (max q used = 17)
- Checking 13-saturation
+ Checking 13-saturation...
Points have successfully been 13-saturated (max q used = 43)
- Checking 17-saturation
+ Checking 17-saturation...
Points have successfully been 17-saturated (max q used = 31)
- Checking 19-saturation
+ Checking 19-saturation...
Points have successfully been 19-saturated (max q used = 37)
done
- P2 = [-2:3:1] is generator number 2
- saturating up to 20...Checking 2-saturation
+ P2 = [-2:3:1] is generator number 2
+ saturating up to 20...Checking 2-saturation...
possible kernel vector = [1,1]
This point may be in 2E(Q): [14:-52:1]
...and it is!
Replacing old generator #1 with new generator [1:-1:1]
Points have successfully been 2-saturated (max q used = 7)
Index gain = 2^1
- Checking 3-saturation
+ Checking 3-saturation...
Points have successfully been 3-saturated (max q used = 13)
- Checking 5-saturation
+ Checking 5-saturation...
Points have successfully been 5-saturated (max q used = 67)
- Checking 7-saturation
+ Checking 7-saturation...
Points have successfully been 7-saturated (max q used = 53)
- Checking 11-saturation
+ Checking 11-saturation...
Points have successfully been 11-saturated (max q used = 73)
- Checking 13-saturation
+ Checking 13-saturation...
Points have successfully been 13-saturated (max q used = 103)
- Checking 17-saturation
+ Checking 17-saturation...
Points have successfully been 17-saturated (max q used = 113)
- Checking 19-saturation
+ Checking 19-saturation...
Points have successfully been 19-saturated (max q used = 47)
done (index = 2).
Gained index 2, new generators = [ [1:-1:1] [-2:3:1] ]
- P3 = [-14:25:8] is generator number 3
- saturating up to 20...Checking 2-saturation
+ P3 = [-14:25:8] is generator number 3
+ saturating up to 20...Checking 2-saturation...
Points have successfully been 2-saturated (max q used = 11)
- Checking 3-saturation
+ Checking 3-saturation...
Points have successfully been 3-saturated (max q used = 13)
- Checking 5-saturation
+ Checking 5-saturation...
Points have successfully been 5-saturated (max q used = 71)
- Checking 7-saturation
+ Checking 7-saturation...
Points have successfully been 7-saturated (max q used = 101)
- Checking 11-saturation
+ Checking 11-saturation...
Points have successfully been 11-saturated (max q used = 127)
- Checking 13-saturation
+ Checking 13-saturation...
Points have successfully been 13-saturated (max q used = 151)
- Checking 17-saturation
+ Checking 17-saturation...
Points have successfully been 17-saturated (max q used = 139)
- Checking 19-saturation
+ Checking 19-saturation...
Points have successfully been 19-saturated (max q used = 179)
done (index = 1).
- P4 = [-1:3:1] = -1*P1 + -1*P2 + -1*P3 (mod torsion)
- P4 = [0:2:1] = 2*P1 + 0*P2 + 1*P3 (mod torsion)
- P4 = [2:13:8] = -3*P1 + 1*P2 + -1*P3 (mod torsion)
- P4 = [1:0:1] = -1*P1 + 0*P2 + 0*P3 (mod torsion)
- P4 = [2:0:1] = -1*P1 + 1*P2 + 0*P3 (mod torsion)
- P4 = [18:7:8] = -2*P1 + -1*P2 + -1*P3 (mod torsion)
- P4 = [3:3:1] = 1*P1 + 0*P2 + 1*P3 (mod torsion)
- P4 = [4:6:1] = 0*P1 + -1*P2 + -1*P3 (mod torsion)
- P4 = [36:69:64] = 1*P1 + -2*P2 + 0*P3 (mod torsion)
- P4 = [68:-25:64] = -2*P1 + -1*P2 + -2*P3 (mod torsion)
- P4 = [12:35:27] = 1*P1 + -1*P2 + -1*P3 (mod torsion)
+ P4 = [-1:3:1] = -1*P1 + -1*P2 + -1*P3 (mod torsion)
+ P4 = [0:2:1] = 2*P1 + 0*P2 + 1*P3 (mod torsion)
+ P4 = [2:13:8] = -3*P1 + 1*P2 + -1*P3 (mod torsion)
+ P4 = [1:0:1] = -1*P1 + 0*P2 + 0*P3 (mod torsion)
+ P4 = [2:0:1] = -1*P1 + 1*P2 + 0*P3 (mod torsion)
+ P4 = [18:7:8] = -2*P1 + -1*P2 + -1*P3 (mod torsion)
+ P4 = [3:3:1] = 1*P1 + 0*P2 + 1*P3 (mod torsion)
+ P4 = [4:6:1] = 0*P1 + -1*P2 + -1*P3 (mod torsion)
+ P4 = [36:69:64] = 1*P1 + -2*P2 + 0*P3 (mod torsion)
+ P4 = [68:-25:64] = -2*P1 + -1*P2 + -2*P3 (mod torsion)
+ P4 = [12:35:27] = 1*P1 + -1*P2 + -1*P3 (mod torsion)
sage: EQ
Subgroup of Mordell-Weil group: [[1:-1:1], [-2:3:1], [-14:25:8]]
@@ -1076,7 +1076,7 @@ class mwrank_MordellWeil(SageObject):
sage: EQ.search(1)
P1 = [0:1:0] is torsion point, order 1
P1 = [-3:0:1] is generator number 1
- saturating up to 20...Checking 2-saturation
+ saturating up to 20...Checking 2-saturation...
...
P4 = [12:35:27] = 1*P1 + -1*P2 + -1*P3 (mod torsion)
sage: EQ
|