The following trees with weighted edges were considered in:
Н. М. Адрианов, А. К. Звонкин
Взвешенные деревья с примитивными группами вращений
рёбер
Фундамент. и прикл. матем., 18:6 (2013), 5–50
http://mi.mathnet.ru/eng/fpm1551
Translated as:
N. Adrianov and A. Zvonkin
Weighted trees with primitive edge rotation groups
Journal of Mathematical Sciences, Vol. 209, No. 2, 2015
http://dx.doi.org/10.1007/s10958-015-2494-2
In this paper these 3 trees are part of the orbit numbered 12.3 of size 6
(the other trees are just mirror images).
These trees have 10 edges of weight 1 and one edge of weight 2. Considered
as maps, each of them has 12 edges. Mathieu group M11 acts on these edges
not by its natural action on 11 points but by its primitive action on 12 points
(for example, on 12 cosets of its subgroup PSL(2,11)).
This text was downloaded from http://logic.pdmi.ras.ru/~yumat/
personaljournal/belyifunction
Let α1 , α2 , α3 be the three solutions of the equation α3 + 1 + ⅈ
11  α2 + 8  0
sol = Solveα ^ 3 + 1 + I * Sqrt[11] * α ^ 2 + 8 == 0 // Simplify;
α1 = α /. sol[[1]]
α1 // N30
1
6
-2 - 2 ⅈ
11 - ⅈ +
22/3 1 + ⅈ
3  5 ⅈ +
3  - 19 + 2 ⅈ
1
11  
11 + 3
- 19 + 2 ⅈ
2
33 - 12 ⅈ
11
11 + 3
33 - 12 ⅈ
11
1/3
-
1/3
- 0.23317703618102519999381368551 - 3.20443184157482742667915844556 ⅈ
α2 = α /. sol[[2]]
α2 // N30
1
6
-2 - 2 ⅈ
11 + - ⅈ +
ⅈ 22/3 ⅈ +
3  5 ⅈ +
3  - 19 + 2 ⅈ
11  
11 + 3
1
2
33 - 12 ⅈ
- 19 + 2 ⅈ
11
11 + 3
33 - 12 ⅈ
11
1/3
0.732345202886266821070455064213 + 1.076053948980735985673733417755 ⅈ
α3 = α /. sol[[3]]
α3 // N30
1
3
-1 - ⅈ
11 + ⅈ 5 ⅈ +
22/3 - 19 + 2 ⅈ
11 + 3
11  
1
2
33 - 12 ⅈ
- 19 + 2 ⅈ
11
11 + 3
33 - 12 ⅈ
11
1/3
+
1/3
- 1.49916816670524162107664137870 - 1.18824689776130840810950770886 ⅈ
Consider the following polynomials :
P0 (α) = -353 739 875 535 087 547 α5 - 2 589 005 798 058 035 158 α4 - 9 879 175 364 331 452 300 α3 32 736 776 927 928 914 060 α2 - 55 105 874 423 315 419 616 α - 55 090 243 936 706 555 995;
P1 (α) = 143 308 931 430 256 794 α5 + 90 908 394 365 818 800 α4 + 833 400 288 653 695 776 α3 1 537 503 720 001 885 512 α2 - 7 473 470 384 863 228 896 α - 15 273 258 442 547 001 624;
P2 (α) = 21 833 021 023 021 329 α5 + 49 707 264 831 199 998 α4 + 244 475 040 567 260 316 α3 +
311 231 213 764 125 204 α2 - 38 464 349 044 108 416 α - 1 002 362 768 723 185 122;
P3 (α) = 786 642 572 955 096 α5 + 2 363 314 995 775 008 α4 + 10 658 113 952 536 512 α3 +
19 729 100 287 160 352 α2 + 15 826 754 491 827 840 α - 16 798 696 373 254 752;
P4 (α) = 9 048 085 615 866 α5 + 24 924 364 792 932 α4 + 115 371 247 474 824 α3 +
193 739 765 454 312 α2 + 115 324 698 591 168 α - 265 982 375 814 441;
P5 (α) = 336 720 835 836 α5 + 456 415 990 272 α4 +
2 824 239 280 896 α3 + 510 803 072 208 α2 - 9 200 899 667 520 α - 22 847 343 943 248;
P6 (α) = 8 106 746 714 α5 + 16 842 482 492 α4 +
86 466 647 608 α3 + 98 069 363 656 α2 - 50 736 667 520 α - 362 701 277 356
1/3
+
2
Orbits_12.nb
P7 (α) = -37 837 800 α5 - 200 962 080 α4 - 781 133 760 α3 - 2 321 551 584 α2 - 3 279 982 464 α - 3 006 371 808;
P8 (α) = -121 407 α5 - 3 460 974 α4 - 10 283 196 α3 - 48 688 860 α2 - 81 716 448 α - 104 571 357;
P9 (α) = -6534 α5 - 14 256 α4 - 85 536 α3 - 116 424 α2 - 142 560 α + 106 920;
P10 (α) = -363 α5 - 858 α4 - 4884 α3 - 7260 α2 - 8448 α + 1518;
P11 (α) = 0;
P12 (α) = 1;
12
A(z) = Pk (α) zk ;
k=0
B(z) = z + 33
For α = α1 , α2 , α3 the ratio A[z] / B[z] is a Belyi function
α1
40
20
0
Out[27]=
-20
-40
-40
0
-20
20
40
2
1
Out[51]=
0
-1
-2
-35
-34
-33
-32
-31
-30
Orbits_12.nb
α2
100
50
0
-50
-100
-60
-40
-20
0
20
40
60
80
α3
40
20
0
-20
-40
-60
-40
-20
0
20
40
60
80
3