This page tabulates the results of a 2013 USRA project that I undertook with the supervision of Kevin Hare and David McKinnon at the University of Waterloo. The goal of the project was to explore the possible directed graph structures of the set of rational preperiodic points of cubic polynomials defined over Q. Using the data from the 2007 Small Height Points for Cubic Polynomials REU project by Rob Benedetto and his students, and by writing a Python program to classify finite functional digraphs by isomorphism, I found there to be exactly 103 distinct preperiodic point portraits exhibited among the first 14 billion conjugacy classes of cubic polynomials ordered by naïve height. My old writeup (from 2013) is available here.
The portraits on this page are grouped into two tables, one for each of Benedetto et al.’s two “normal forms” for cubic polynomials:
Every cubic polynomial over Q is linearly conjugate (viz. via some z ↦ αz + β) to one in normal form (cf. §4 of their paper). Since conjugacy induces an isomorphism between associated functional digraphs, the preperiodic point portrait of a cubic polynomial depends only on its normal form.
The portraits are depicted as unlabelled directed graphs (hand-coded in TikZ). Each portrait Γ is accompanied by a listing of several statistics, namely:
There are 79 portraits in Table 1 and 37 portraits in Table 2. The two tables have 13 portraits in common: portraits 1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 17, 29, and 33 in Table 2 are isomorphic to portraits 29, 30, 3, 4, 40, 14, 44, 53, 49, 43, 70, 66, and 33 in Table 1, respectively.
The tables can be sorted ascending or descending by clicking on the column headers.
# | Γ | N | f(z) | H | H* | p | c | l | k | cycles | degree | structure |
---|---|---|---|---|---|---|---|---|---|---|---|---|
1. | 11,939,559,973 (21 MB) | z3 + 1 | 1 | 1 | 0 | 0 | 0 | 0 | (0, 0, 0, 0) | |||
2. | 84,404 (3 MB) | −z3 + 1 | 1 | 1 | 2 | 2 | 2 | 1 | 2 | (0, 2, 0, 0) | 2m: 2 | |
3. | 282,083 (12 MB) | z3 + z + 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | (0, 1, 0, 0) | 1m: 1 | |
4. | 112,291 (6 MB) | z3 − z + 1 | 1 | 1 | 3 | 2 | 1 | 1 | 1 | (2, 0, 0, 1) | 1m: 1, 2 | |
5. | 5 (351 B) | z3/2 − z/2 + 1 | 2 | 2 | 4 | 2 | 1 | 2 | 1, 1 | (2, 1, 0, 1) | 1m: 2, 2 | |
6. | 1,003 (78 kB) | −2z3 + z/2 + 1 | 2 | 4 | 4 | 3 | 2 | 1 | 2 | (2, 1, 0, 1) | 2m: 2, 2 | |
7. | 91 (5 kB) | −z3 + 3z + 1 | 3 | 3 | 2 | 2 | 1 | 1 | 1 | (1, 0, 1, 0) | 1m: 1, 1 | |
8. | 2 (149 B) | z3/2 − 3z/2 + 1 | 3 | 3 | 5 | 3 | 2 | 2 | 1, 2 | (2, 1, 2, 0) | 1m: 1, 1 2m: 2, 1 |
|
9. | 1 (67 B) | −z3/2 + 3z/2 + 1 | 3 | 3 | 5 | 4 | 3 | 1 | 3 | (2, 1, 2, 0) | 3m: 3, 2 | |
10. | 277 (24 kB) | z3/3 − z/3 + 1 | 3 | 3 | 4 | 3 | 1 | 1 | 1 | (2, 1, 0, 1) | 1m: 1, 2, 1 | |
11. | 42 (2 kB) | 4z3 − 2z + 1 | 4 | 4 | 2 | 1 | 1 | 2 | 1, 1 | (0, 2, 0, 0) | 1m: 2 | |
12. | 7 (420 B) | −4z3 + 3z + 1 | 4 | 4 | 3 | 3 | 2 | 1 | 2 | (1, 1, 1, 0) | 2m: 2, 1 | |
13. | 57 (5 kB) | z3/3 − 4z/3 + 1 | 4 | 4 | 5 | 4 | 2 | 1 | 2 | (2, 2, 0, 1) | 2m: 2, 2, 1 | |
14. | 274 (25 kB) | 4z3/3 − 4z/3 + 1 | 4 | 4 | 5 | 2 | 1 | 3 | 1, 1, 1 | (2, 2, 0, 1) | 1m: 3, 2 | |
15. | 80 (6 kB) | −z3 + 5z/4 + 1 | 5 | 5 | 3 | 3 | 3 | 1 | 3 | (0, 3, 0, 0) | 3m: 3 | |
16. | 7 (552 B) | −2z3/3 + 5z/3 + 1 | 5 | 5 | 4 | 4 | 4 | 1 | 4 | (0, 4, 0, 0) | 4m: 4 | |
17. | 29 (3 kB) | z3/6 − z/6 + 1 | 6 | 6 | 6 | 3 | 1 | 3 | 1, 1, 1 | (2, 3, 0, 1) | 1m: 3, 2, 1 | |
18. | 14 (1 kB) | −z3/6 + z/6 + 1 | 6 | 6 | 5 | 4 | 1 | 1 | 1 | (2, 2, 0, 1) | 1m: 1, 2, 1, 1 | |
19. | 12 (1 kB) | −2z3/3 + z/6 + 1 | 6 | 6 | 6 | 3 | 2 | 2 | 2, 2 | (2, 3, 0, 1) | 2m: 4, 2 | |
20. | 21 (2 kB) | −3z3/2 + z/6 + 1 | 6 | 9 | 5 | 3 | 2 | 2 | 1, 2 | (2, 2, 0, 1) | 1m: 1 2m: 2, 2 |
|
21. | 6 (595 B) | z3/3 − 7z/3 + 1 | 7 | 7 | 7 | 4 | 1 | 1 | 1 | (4, 1, 0, 2) | 1m: 1, 2, 3, 1 | |
22. | 10 (1 kB) | 2z3/3 − 7z/6 + 1 | 7 | 7 | 8 | 3 | 1 | 3 | 1, 1, 1 | (4, 2, 0, 2) | 1m: 3, 4, 1 | |
23. | 13 (1 kB) | −2z3/3 + 7z/6 + 1 | 7 | 7 | 7 | 4 | 2 | 1 | 2 | (4, 1, 0, 2) | 2m: 2, 4, 1 | |
24. | 1 (112 B) | −3z3/2 + 7z/6 + 1 | 7 | 9 | 8 | 5 | 2 | 1 | 2 | (4, 2, 0, 2) | 2m: 2, 2, 3, 1 | |
25. | 2 (201 B) | z3/6 − 7z/6 + 1 | 7 | 7 | 7 | 4 | 2 | 1 | 2 | (4, 1, 0, 2) | 2m: 2, 2, 3 | |
26. | 2 (182 B) | −z3/6 + 7z/6 + 1 | 7 | 7 | 7 | 4 | 1 | 1 | 1 | (4, 1, 0, 2) | 1m: 1, 2, 1, 3 | |
27. | 2 (115 B) | 3z3/2 − 9z/2 + 1 | 9 | 9 | 3 | 3 | 1 | 1 | 1 | (1, 1, 1, 0) | 1m: 1, 1, 1 | |
28. | 8 (1 kB) | z3/4 − 9z/4 + 1 | 9 | 9 | 5 | 4 | 3 | 1 | 3 | (2, 2, 0, 1) | 3m: 3, 2 | |
29. | 840 (59 kB) | 9z3/2 − 5z/2 + 1 | 9 | 9 | 3 | 1 | 1 | 3 | 1, 1, 1 | (0, 3, 0, 0) | 1m: 3 | |
30. | 210 (12 kB) | −9z3/10 − z/10 + 1 | 10 | 10 | 3 | 2 | 2 | 2 | 1, 2 | (0, 3, 0, 0) | 1m: 1 2m: 2 |
|
31. | 80 (5 kB) | −9z3/10 + 11z/10 + 1 | 11 | 11 | 4 | 2 | 2 | 2 | 2, 2 | (0, 4, 0, 0) | 2m: 4 | |
32. | 2 (216 B) | z3/6 − 13z/6 + 1 | 13 | 13 | 9 | 5 | 2 | 1 | 2 | (4, 3, 0, 2) | 2m: 2, 4, 2, 1 | |
33. | 1 (108 B) | −z3/6 + 13z/6 + 1 | 13 | 13 | 9 | 3 | 1 | 3 | 1, 1, 1 | (4, 3, 0, 2) | 1m: 3, 4, 2 | |
34. | 1 (106 B) | 2z3/3 − 13z/6 + 1 | 13 | 13 | 8 | 5 | 1 | 1 | 1 | (4, 2, 0, 2) | 1m: 1, 2, 1, 1, 3 | |
35. | 1 (110 B) | −4z3/3 + 13z/12 + 1 | 13 | 16 | 8 | 5 | 3 | 1 | 3 | (4, 2, 0, 2) | 3m: 3, 4, 1 | |
36. | 92 (10 kB) | z3/16 − 7z/4 + 1 −9z3/10 + 21z/10 + 1 |
16 | 21 | 6 | 3 | 2 | 1 | 2 | (4, 0, 0, 2) | 2m: 2, 4 | |
37. | 1 (89 B) | −z3/16 + 7z/4 + 1 | 16 | 28 | 6 | 2 | 1 | 2 | 1, 1 | (4, 0, 0, 2) | 1m: 2, 4 | |
38. | 51 (5 kB) | 3z3/16 − 7z/12 + 1 2z3/15 − 38z/15 + 1 |
16 | 38 | 6 | 3 | 1 | 1 | 1 | (4, 0, 0, 2) | 1m: 1, 2, 3 | |
39. | 2 (133 B) | 3z3/16 − 9z/4 + 1 | 16 | 36 | 4 | 3 | 1 | 1 | 1 | (2, 0, 2, 0) | 1m: 1, 1, 2 | |
40. | 1 (88 B) | 9z3/16 − 3z/4 + 1 | 16 | 16 | 5 | 2 | 1 | 3 | 1, 1, 1 | (2, 1, 2, 0) | 1m: 3, 2 | |
41. | 1 (78 B) | −9z3/16 + 3z/4 + 1 | 16 | 16 | 4 | 3 | 2 | 1 | 2 | (2, 0, 2, 0) | 2m: 2, 2 | |
42. | 1 (86 B) | −16z3/15 + 16z/15 + 1 | 16 | 16 | 6 | 3 | 2 | 2 | 1, 2 | (2, 3, 0, 1) | 1m: 1, 2, 1 2m: 2 |
|
43. | 88 (10 kB) | 9z3/10 − 21z/10 + 1 | 21 | 21 | 7 | 2 | 1 | 3 | 1, 1, 1 | (4, 1, 0, 2) | 1m: 3, 4 | |
44. | 11 (1 kB) | 25z3/12 − 25z/12 + 1 | 25 | 25 | 5 | 2 | 2 | 2 | 1, 2 | (2, 2, 0, 1) | 1m: 1, 2 2m: 2 |
|
45. | 1 (165 B) | z3/12 − 25z/12 + 1 | 25 | 25 | 7 | 6 | 5 | 1 | 5 | (2, 4, 0, 1) | 5m: 5, 2 | |
46. | 1 (89 B) | −z3/12 + 25z/12 + 1 | 25 | 25 | 8 | 4 | 3 | 4 | 1, 1, 1, 3 | (2, 5, 0, 1) | 1m: 3 3m: 3, 2 |
|
47. | 1 (90 B) | 3z3/4 − 25z/12 + 1 | 25 | 25 | 7 | 5 | 3 | 2 | 1, 3 | (2, 4, 0, 1) | 1m: 1 3m: 3, 2, 1 |
|
48. | 4 (331 B) | 7z3/6 − 25z/6 + 1 | 25 | 25 | 4 | 3 | 3 | 2 | 1, 3 | (0, 4, 0, 0) | 1m: 1 3m: 3 |
|
49. | 11 (1 kB) | −2z3/15 + 19z/30 + 1 | 30 | 30 | 5 | 3 | 1 | 1 | 1 | (2, 2, 0, 1) | 1m: 1, 2, 2 | |
50. | 1 (105 B) | −z3/3 + 37z/12 + 1 | 37 | 37 | 7 | 5 | 2 | 2 | 1, 2 | (2, 4, 0, 1) | 1m: 1 2m: 2, 2, 1, 1 |
|
51. | 1 (101 B) | −4z3/3 + 37z/12 + 1 | 37 | 37 | 6 | 5 | 1 | 1 | 1 | (2, 3, 0, 1) | 1m: 1, 2, 1, 1, 1 | |
52. | 2 (368 B) | −32z3/3 + 37z/6 + 1 | 37 | 64 | 6 | 5 | 2 | 1 | 2 | (2, 3, 0, 1) | 2m: 2, 2, 1, 1 | |
53. | 2 (240 B) | 32z3/15 − 47z/15 + 1 | 47 | 47 | 5 | 2 | 2 | 4 | 1, 1, 1, 2 | (0, 5, 0, 0) | 1m: 3 2m: 2 |
|
54. | 2 (223 B) | z3/48 − 19z/12 + 1 | 48 | 76 | 8 | 5 | 2 | 1 | 2 | (4, 2, 0, 2) | 2m: 2, 4, 1, 1 | |
55. | 2 (230 B) | z3/48 − 31z/12 + 1 −49z3/48 + 19z/12 + 1 |
48 | 76 | 9 | 4 | 2 | 2 | 2, 2 | (4, 3, 0, 2) | 2m: 4, 4, 1 | |
56. | 1 (118 B) | 25z3/24 − 49z/24 + 1 | 49 | 49 | 9 | 4 | 2 | 1 | 2 | (4, 3, 0, 2) | 2m: 2, 4, 3 | |
57. | 1 (131 B) | 5z3/12 − 49z/60 + 1 | 60 | 60 | 9 | 4 | 1 | 1 | 1 | (6, 0, 0, 3) | 1m: 1, 2, 3, 3 | |
58. | 3 (394 B) | −5z3/12 + 49z/60 + 1 | 60 | 60 | 9 | 4 | 2 | 1 | 2 | (6, 0, 0, 3) | 2m: 2, 4, 3 | |
59. | 1 (77 B) | 6z3/5 − 61z/30 + 1 | 61 | 61 | 6 | 5 | 4 | 1 | 4 | (2, 3, 0, 1) | 4m: 4, 2 | |
60. | 1 (118 B) | z3/6 − 73z/24 + 1 | 73 | 73 | 8 | 4 | 2 | 1 | 2 | (4, 2, 0, 2) | 2m: 2, 2, 4 | |
61. | 2 (465 B) | −z3/6 + 73z/24 + 1 | 73 | 73 | 6 | 4 | 1 | 1 | 1 | (2, 3, 0, 1) | 1m: 1, 2, 2, 1 | |
62. | 1 (84 B) | z3/30 − 79z/30 + 1 | 79 | 79 | 7 | 4 | 2 | 2 | 2, 2 | (2, 4, 0, 1) | 2m: 4, 2, 1 | |
63. | 3 (348 B) | 49z3/30 − 79z/30 + 1 | 79 | 79 | 6 | 4 | 2 | 2 | 1, 2 | (2, 3, 0, 1) | 1m: 1 2m: 2, 2, 1 |
|
64. | 2 (303 B) | −z3/30 + 91z/30 + 1 | 91 | 91 | 8 | 5 | 3 | 1 | 3 | (4, 2, 0, 2) | 3m: 3, 4, 1 | |
65. | 1 (86 B) | 8z3/15 − 121z/30 + 1 | 121 | 121 | 6 | 5 | 3 | 1 | 3 | (2, 3, 0, 1) | 3m: 3, 2, 1 | |
66. | 4 (683 B) | 121z3/80 − 91z/20 + 1 | 121 | 364 | 9 | 2 | 1 | 3 | 1, 1, 1 | (6, 0, 0, 3) | 1m: 3, 6 | |
67. | 1 (94 B) | −121z3/80 + 71z/20 + 1 | 121 | 284 | 6 | 2 | 2 | 3 | 2, 2, 2 | (0, 6, 0, 0) | 2m: 6 | |
68. | 5 (373 B) | 18z3/125 − 3z/10 + 1 144z3 − 12z + 1 |
125 | 144 | 4 | 2 | 1 | 3 | 1, 1, 1 | (1, 2, 1, 0) | 1m: 3, 1 | |
69. | 3 (215 B) | 16z3/125 − 12z/5 + 1 −121z3/250 + 3z/10 + 1 |
125 | 250 | 4 | 2 | 2 | 2 | 1, 2 | (1, 2, 1, 0) | 1m: 1, 1 2m: 2 |
|
70. | 1 (164 B) | 3z3/128 − 55z/24 + 1 | 128 | 880 | 5 | 4 | 4 | 2 | 1, 4 | (0, 5, 0, 0) | 1m: 1 4m: 4 |
|
71. | 3 (484 B) | −147z3/80 + 31z/60 + 1 | 147 | 441 | 8 | 3 | 2 | 2 | 2, 2 | (4, 2, 0, 2) | 2m: 4, 4 | |
72. | 1 (95 B) | −z3/120 + 169z/120 + 1 | 169 | 169 | 7 | 4 | 1 | 3 | 1, 1, 1 | (2, 4, 0, 1) | 1m: 3, 2, 1, 1 | |
73. | 1 (115 B) | 49z3/120 − 169z/120 + 1 | 169 | 169 | 8 | 4 | 2 | 2 | 1, 2 | (4, 2, 0, 2) | 1m: 1 2m: 2, 4, 1 |
|
74. | 1 (130 B) | −169z3/96 + 109z/24 + 1 | 169 | 436 | 8 | 3 | 2 | 3 | 2, 2, 2 | (2, 5, 0, 1) | 2m: 6, 2 | |
75. | 1 (214 B) | −147z3/10 + 181z/30 + 1 | 181 | 441 | 7 | 3 | 2 | 4 | 1, 1, 1, 2 | (2, 4, 0, 1) | 1m: 3 2m: 2, 2 |
|
76. | 1 (120 B) | z3/240 − 151z/60 + 1 | 240 | 604 | 10 | 3 | 2 | 3 | 2, 2, 2 | (4, 4, 0, 2) | 2m: 6, 4 | |
77. | 1 (99 B) | 243z3/10 − 151z/30 + 1 | 243 | 729 | 6 | 4 | 3 | 2 | 1, 3 | (2, 3, 0, 1) | 1m: 1 3m: 3, 2 |
|
78. | 2 (163 B) | −49z3/250 + 27z/10 + 1 −289z3/16 + 27z/4 + 1 |
250 | 289 | 4 | 4 | 2 | 1 | 2 | (1, 2, 1, 0) | 2m: 2, 1, 1 | |
79. | 1 (119 B) | 27z3/256 − 133z/48 + 1 | 256 | 2128 | 7 | 4 | 3 | 1 | 3 | (4, 1, 0, 2) | 3m: 3, 4 |
# | Γ | N | f(z) | H | p | c | l | k | cycles | degree | structure |
---|---|---|---|---|---|---|---|---|---|---|---|
1. | 94,459 (5 MB) | z3 | 1 | 3 | 1 | 1 | 3 | 1, 1, 1 | (0, 3, 0, 0) | 1m: 3 | |
2. | 94,459 (4 MB) | −z3 | 1 | 3 | 2 | 2 | 2 | 1, 2 | (0, 3, 0, 0) | 1m: 1 2m: 2 |
|
3. | 2,072,790,590 (32 kB) | z3 + z | 1 | 1 | 1 | 1 | 1 | 1 | (0, 1, 0, 0) | 1m: 1 | |
4. | 108,794 (6 MB) | z3 − z | 1 | 3 | 2 | 1 | 1 | 1 | (2, 0, 0, 1) | 1m: 1, 2 | |
5. | 2 (133 B) | z3 − 3z | 3 | 5 | 2 | 1 | 3 | 1, 1, 1 | (2, 1, 2, 0) | 1m: 3, 2 | |
6. | 2 (125 B) | −z3 + 3z | 3 | 5 | 3 | 2 | 2 | 1, 2 | (2, 1, 2, 0) | 1m: 1 2m: 2, 2 |
|
7. | 201 (15 kB) | 3z3 − z/3 | 3 | 5 | 2 | 1 | 3 | 1, 1, 1 | (2, 2, 0, 1) | 1m: 3, 2 | |
8. | 201 (14 kB) | −3z3 + z/3 | 3 | 5 | 2 | 2 | 2 | 1, 2 | (2, 2, 0, 1) | 1m: 1, 2 2m: 2 |
|
9. | 174 (13 kB) | z3 − 5z/4 | 5 | 5 | 2 | 2 | 4 | 1, 1, 1, 2 | (0, 5, 0, 0) | 1m: 3 2m: 2 |
|
10. | 116 (6 kB) | 2z3 − 5z/2 | 5 | 5 | 2 | 2 | 3 | 1, 2, 2 | (0, 5, 0, 0) | 1m: 1 2m: 4 |
|
11. | 46 (3 kB) | 3z3/2 − z/6 | 6 | 5 | 3 | 1 | 1 | 1 | (2, 2, 0, 1) | 1m: 1, 2, 2 | |
12. | 149 (16 kB) | 2z3 − 7z/2 | 7 | 7 | 2 | 1 | 3 | 1, 1, 1 | (4, 1, 0, 2) | 1m: 3, 4 | |
13. | 149 (16 kB) | −2z3 + 7z/2 | 7 | 7 | 3 | 2 | 2 | 1, 2 | (4, 1, 0, 2) | 1m: 1 2m: 2, 4 |
|
14. | 8 (878 B) | 3z3/2 − 13z/6 | 13 | 9 | 3 | 2 | 3 | 1, 2, 2 | (4, 3, 0, 2) | 1m: 1 2m: 4, 4 |
|
15. | 2 (254 B) | 3z3 − 13z/12 | 13 | 9 | 3 | 2 | 4 | 1, 1, 1, 2 | (4, 3, 0, 2) | 1m: 3 2m: 2, 4 |
|
16. | 2 (254 B) | −3z3 + 13z/12 | 13 | 9 | 2 | 2 | 4 | 1, 1, 1, 2 | (4, 3, 0, 2) | 1m: 3, 4 2m: 2 |
|
17. | 8 (444 B) | 6z3/5 − 17z/6 | 17 | 5 | 4 | 4 | 2 | 1, 4 | (0, 5, 0, 0) | 1m: 1 4m: 4 |
|
18. | 1 (139 B) | 3z3/2 − 19z/6 | 19 | 11 | 3 | 1 | 3 | 1, 1, 1 | (4, 5, 0, 2) | 1m: 3, 4, 4 | |
19. | 1 (136 B) | −3z3/2 + 19z/6 | 19 | 11 | 4 | 2 | 2 | 1, 2 | (4, 5, 0, 2) | 1m: 1 2m: 2, 4, 4 |
|
20. | 3 (280 B) | 3z3/2 − z/24 | 24 | 7 | 3 | 1 | 3 | 1, 1, 1 | (2, 4, 0, 1) | 1m: 3, 2, 2 | |
21. | 3 (269 B) | −3z3/2 + z/24 | 24 | 7 | 3 | 2 | 2 | 1, 2 | (2, 4, 0, 1) | 1m: 1, 2, 2 2m: 2 |
|
22. | 4 (706 B) | 3z3 − 25z/12 | 25 | 7 | 2 | 2 | 3 | 1, 2, 2 | (2, 4, 0, 1) | 1m: 1, 2 2m: 4 |
|
23. | 4 (370 B) | 3z3/2 − 25z/24 | 25 | 7 | 2 | 2 | 4 | 1, 1, 1, 2 | (2, 4, 0, 1) | 1m: 3, 2 2m: 2 |
|
24. | 5 (401 B) | 5z3/3 − 34z/15 | 34 | 7 | 2 | 2 | 5 | 1, 1, 1, 2, 2 | (0, 7, 0, 0) | 1m: 3 2m: 4 |
|
25. | 5 (382 B) | −5z3/3 + 34z/15 | 34 | 7 | 2 | 2 | 4 | 1, 2, 2, 2 | (0, 7, 0, 0) | 1m: 1 2m: 6 |
|
26. | 2 (284 B) | 3z3 − 37z/12 | 37 | 11 | 3 | 2 | 4 | 1, 1, 1, 2 | (4, 5, 0, 2) | 1m: 3, 4, 2 2m: 2 |
|
27. | 2 (284 B) | −3z3 + 37z/12 | 37 | 11 | 4 | 2 | 4 | 1, 1, 1, 2 | (4, 5, 0, 2) | 1m: 3 2m: 2, 4, 2 |
|
28. | 5 (637 B) | 3z3/2 − 49z/24 | 49 | 9 | 3 | 2 | 2 | 1, 2 | (6, 0, 0, 3) | 1m: 1, 2 2m: 2, 4 |
|
29. | 5 (658 B) | −3z3/2 + 49z/24 | 49 | 9 | 2 | 1 | 3 | 1, 1, 1 | (6, 0, 0, 3) | 1m: 3, 6 | |
30. | 1 (132 B) | 3z3/2 − 73z/24 | 73 | 11 | 3 | 2 | 4 | 1, 2, 2, 2 | (4, 5, 0, 2) | 1m: 1 2m: 6, 4 |
|
31. | 1 (137 B) | −3z3/2 + 73z/24 | 73 | 11 | 3 | 2 | 5 | 1, 1, 1, 2, 2 | (4, 5, 0, 2) | 1m: 3 2m: 4, 4 |
|
32. | 2 (256 B) | 7z3/6 − 163z/42 | 163 | 9 | 4 | 2 | 2 | 1, 2 | (4, 3, 0, 2) | 1m: 1 2m: 2, 4, 2 |
|
33. | 2 (266 B) | −7z3/6 + 163z/42 | 163 | 9 | 3 | 1 | 3 | 1, 1, 1 | (4, 3, 0, 2) | 1m: 3, 4, 2 | |
34. | 1 (160 B) | 6z3/5 − 169z/120 | 169 | 11 | 3 | 2 | 4 | 1, 1, 1, 2 | (6, 2, 0, 3) | 1m: 3, 2 2m: 2, 4 |
|
35. | 1 (161 B) | −6z3/5 + 169z/120 | 169 | 11 | 2 | 2 | 4 | 1, 1, 1, 2 | (6, 2, 0, 3) | 1m: 3, 6 2m: 2 |
|
36. | 1 (111 B) | 6z3/5 − 289z/120 | 289 | 9 | 2 | 2 | 4 | 1, 2, 2, 2 | (2, 6, 0, 1) | 1m: 1, 2 2m: 6 |
|
37. | 1 (116 B) | −6z3/5 + 289z/120 | 289 | 9 | 2 | 2 | 5 | 1, 1, 1, 2, 2 | (2, 6, 0, 1) | 1m: 3, 2 2m: 4 |
Published: 28 Mar 2021Last modified: 28 Mar 2021