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1. sage: E = EllipticCurve( GF(13), [3, 8] )
2. sage: E
3. Elliptic Curve defined by y^2 = x^3 + 3*x + 8 over Finite Field of size 13
4. sage: E.order().factor()
5. 3^2
6. sage: for P in E.points():
7. ....:     print "Point %8s of order %s" % (P.xy() if P != E(0) else 'ZERO', P.order())
8. ....:
9. Point     ZERO of order 1
10. Point   (1, 5) of order 9
11. Point   (1, 8) of order 9
12. Point   (2, 3) of order 9
13. Point  (2, 10) of order 9
14. Point   (9, 6) of order 3
15. Point   (9, 7) of order 3
16. Point  (12, 2) of order 9
17. Point (12, 11) of order 9
18. sage: E.gens()
19. ((12 : 11 : 1),)
20. sage: G = E.gens()[0]    # the generator
21. sage: G.order()
22. 9
23.
24. sage: E.division_polynomial(3).factor()
25. (3) * (x + 4) * (x^3 + 9*x^2 + 9*x + 9)
26.
27. sage: F.<a> = GF( 13^3, modulus = x^3 + 9*x^2 + 9*x + 9 )
28. sage: F
29. Finite Field in a of size 13^3
30. sage: F.modulus()
31. x^3 + 9*x^2 + 9*x + 9
32. sage: a.minpoly()
33. x^3 + 9*x^2 + 9*x + 9
34. sage: # we consider the base change of E from GF(13) to F = GF(13^3)
35. sage: EF = E.base_extend(F)
36. sage: EF
37. Elliptic Curve defined by y^2 = x^3 + 3*x + 8 over Finite Field in a of size 13^3
38. sage: EF.base_field()
39. Finite Field in a of size 13^3
40. sage: EF.base_field() == F
41. True
42.
43. sage: EF.order().factor()
44. 2^2 * 3^4 * 7
45.
46. sage: for P in EF.points():
47. ....:     if P.order() == 3:
48. ....:         print P
49. ....:
50. (9 : 6 : 1)
51. (9 : 7 : 1)
52. (a : 2*a + 5 : 1)
53. (a : 11*a + 8 : 1)
54. (4*a^2 + 3*a : 5*a^2 + 7*a + 8 : 1)
55. (4*a^2 + 3*a : 8*a^2 + 6*a + 5 : 1)
56. (9*a^2 + 9*a + 4 : 5*a^2 + 5*a : 1)
57. (9*a^2 + 9*a + 4 : 8*a^2 + 8*a : 1)
58. sage:
59.
60. sage: for f, multiplicity in E.division_polynomial(9).factor():
61. ....:     print f
62. ....:
63. x + 1
64. x + 4
65. x + 11
66. x + 12
67. x^3 + 4*x^2 + 8*x + 7
68. x^3 + 8*x^2 + x + 6
69. x^3 + 9*x^2 + 9*x + 9
70. x^9 + 2*x^7 + 5*x^6 + 11*x^5 + x^4 + 12*x^3 + 11*x^2 + 7*x + 9
71. x^9 + 6*x^8 + 7*x^7 + x^6 + x^5 + 10*x^4 + 9*x^3 + 2*x^2 + 7*x + 7
72. x^9 + 10*x^8 + 6*x^7 + 7*x^6 + 3*x^5 + 3*x^4 + 7*x^3 + 9*x^2 + 7*x + 10
73.
74. sage: EL = E.base_extend(L)
75. sage: for bb in b.minpoly().roots(ring=L, multiplicities=False):
76. ....:     ybb = sqrt( bb^3 + 3*bb + 8 )
77. ....:     P = EL.point( (bb, ybb) )
78. ....:     print "Order %s for point with components:n%sn%sn" % (P.order(), P.xy()[0], P.xy()[1])
79. ....:
80. Order 9 for point with components:
81. b
82. 11*b^8 + 7*b^7 + 5*b^6 + 2*b^4 + 10*b^3 + 7*b^2 + 6*b + 11
83.
84. Order 9 for point with components:
85. 3*b^8 + 4*b^7 + 4*b^6 + 3*b^5 + b^4 + 9*b^3 + 9*b
86. 8*b^8 + 12*b^7 + 5*b^6 + 11*b^5 + 9*b^4 + 2*b^3 + 2*b^2 + 12*b + 12
87.
88. Order 9 for point with components:
89. 11*b^8 + 5*b^7 + 7*b^6 + 2*b^5 + 7*b^4 + 7*b^2 + b + 12
90. 2*b^8 + 6*b^7 + 2*b^6 + 7*b^5 + 5*b^4 + 6*b^3 + 5*b^2 + 11
91.
92. Order 9 for point with components:
93. 4*b^8 + 9*b^7 + 4*b^5 + b^4 + 6*b^3 + b^2 + 11
94. b^8 + b^7 + 5*b^6 + b^5 + 12*b^4 + 2*b^2 + 5*b + 7
95.
96. Order 9 for point with components:
97. 6*b^8 + 10*b^7 + 3*b^6 + 8*b^5 + 2*b^4 + 10*b^3 + 3*b + 11
98. 10*b^8 + 5*b^7 + 3*b^6 + 3*b^5 + 3*b^4 + 7*b^3 + b^2 + 7*b + 11
99.
100. Order 9 for point with components:
101. 5*b^8 + 6*b^7 + b^6 + 6*b^5 + 8*b^4 + 2*b^2 + 9*b + 10
102. 7*b^8 + 3*b^7 + 6*b^6 + b^5 + 11*b^4 + 9*b^3 + 2*b^2 + 5*b + 4
103.
104. Order 9 for point with components:
105. 8*b^8 + 10*b^7 + 5*b^5 + 6*b^3 + 12*b^2 + b + 9
106. 2*b^7 + 2*b^6 + 3*b^5 + 5*b^4 + 12*b^3 + 10*b^2 + 6*b + 5
107.
108. Order 9 for point with components:
109. 6*b^8 + b^7 + 7*b^6 + 3*b^5 + 6*b^4 + 2*b^3 + 10*b + 6
110. 11*b^8 + 9*b^7 + 4*b^6 + 4*b^5 + 12*b^4 + 5*b + 10
111.
112. Order 9 for point with components:
113. 9*b^8 + 7*b^7 + 4*b^6 + 8*b^5 + b^4 + 6*b^3 + 4*b^2 + 5*b + 6
114. 2*b^8 + 5*b^7 + b^6 + 5*b^5 + 7*b^4 + 6*b^3 + 4*b + 5
115.
116. sage:
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