Passport invariants
| Degree: | $8$ |
| Monodromy group: | $S_8$ |
| Genus: | $0$ |
| Geometry type: | hyperbolic |
| Primitive: | yes |
Conjugacy class data
The order and cycle type of an element in each of the conjugacy classes $C_0, C_1, C_{\infty}$ of the passport containing this orbit.
| Order | Partition |
| $6$ | $6, 1, 1$ |
| $6$ | $3, 3, 2$ |
| $6$ | $3, 2, 2, 1$ |
Base field
6.4.285768000.2 ; Generator \(\nu\), with minimal polynomial \( T^{6} - 3 T^{4} - 6 T^{3} - 12 T^{2} + 24 T - 8 \).
Curve
| $\mathbb{P}^1$, with affine coordinate $x$ | |
|
$\displaystyle \left(1/2 \left(-4 \nu^{5} - 7 \nu^{4} + 17 \nu^{3} + 22 \nu^{2} + 106 \nu - 90\right)\right) x^{6} \left(1/4 \left(7 \nu^{5} - 28 \nu^{4} - 19 \nu^{3} + 148 \nu^{2} - 68 \nu + 12\right) x^{2} + \left(28 \nu^{5} + 20 \nu^{4} - 122 \nu^{3} - 176 \nu^{2} - 206 \nu + 152\right) x - 31 \nu^{5} - 27 \nu^{4} + 73 \nu^{3} + 267 \nu^{2} + 572 \nu - 346\right) t + \left(1/4 \left(-407 \nu^{5} - 1000 \nu^{4} - 1235 \nu^{3} - 590 \nu^{2} + 3436 \nu - 1324\right)\right) \left(\left(4 \nu^{5} + 2 \nu^{4} - 11 \nu^{3} - 29 \nu^{2} - 62 \nu + 66\right) x + 1/4 \left(13 \nu^{5} + 6 \nu^{4} - 37 \nu^{3} - 94 \nu^{2} - 200 \nu + 228\right)\right) \left(1/4 \left(7 \nu^{5} + 6 \nu^{4} - 23 \nu^{3} - 46 \nu^{2} - 112 \nu + 52\right) x^{2} + 1/4 \left(-39 \nu^{5} - 32 \nu^{4} + 97 \nu^{3} + 310 \nu^{2} + 716 \nu - 416\right) x + 1/4 \left(47 \nu^{5} + 36 \nu^{4} - 115 \nu^{3} - 368 \nu^{2} - 848 \nu + 496\right)\right)^{2}=0$
|
Map
\(\displaystyle \phi(x) =\)
$\displaystyle \frac{3125}{1411243} \frac{\left(13\!\cdots\!12 \nu^{5} + 65\!\cdots\!20 \nu^{4} - 37\!\cdots\!04 \nu^{3} - 98\!\cdots\!04 \nu^{2} - 20\!\cdots\!36 \nu + 22\!\cdots\!06\right) x^{8} + \left(-14\!\cdots\!02 \nu^{5} - 71\!\cdots\!00 \nu^{4} + 40\!\cdots\!94 \nu^{3} + 10\!\cdots\!44 \nu^{2} + 22\!\cdots\!56 \nu - 24\!\cdots\!16\right) x^{7} + \left(16\!\cdots\!75 \nu^{5} + 78\!\cdots\!47 \nu^{4} - 44\!\cdots\!23 \nu^{3} - 11\!\cdots\!49 \nu^{2} - 25\!\cdots\!22 \nu + 26\!\cdots\!52\right) x^{6}}{13\!\cdots\!50 x^{8} + \left(-23\!\cdots\!50 \nu^{5} - 11\!\cdots\!00 \nu^{4} + 65\!\cdots\!50 \nu^{3} + 17\!\cdots\!00 \nu^{2} + 37\!\cdots\!00 \nu - 38\!\cdots\!00\right) x^{7} + \left(91\!\cdots\!65 \nu^{5} + 44\!\cdots\!25 \nu^{4} - 25\!\cdots\!25 \nu^{3} - 67\!\cdots\!75 \nu^{2} - 14\!\cdots\!50 \nu + 15\!\cdots\!20\right) x^{6} + \left(-11\!\cdots\!44 \nu^{5} - 57\!\cdots\!72 \nu^{4} + 32\!\cdots\!76 \nu^{3} + 86\!\cdots\!12 \nu^{2} + 18\!\cdots\!04 \nu - 19\!\cdots\!64\right) x^{5} + \left(13\!\cdots\!30 \nu^{5} + 63\!\cdots\!40 \nu^{4} - 36\!\cdots\!70 \nu^{3} - 95\!\cdots\!40 \nu^{2} - 20\!\cdots\!80 \nu + 21\!\cdots\!80\right) x^{4} + \left(-14\!\cdots\!40 \nu^{5} - 68\!\cdots\!20 \nu^{4} + 38\!\cdots\!60 \nu^{3} + 10\!\cdots\!20 \nu^{2} + 21\!\cdots\!40 \nu - 23\!\cdots\!40\right) x^{3} + \left(-11\!\cdots\!65 \nu^{5} - 57\!\cdots\!95 \nu^{4} + 32\!\cdots\!85 \nu^{3} + 86\!\cdots\!45 \nu^{2} + 18\!\cdots\!90 \nu - 19\!\cdots\!40\right) x^{2} + \left(13\!\cdots\!70 \nu^{5} + 63\!\cdots\!40 \nu^{4} - 36\!\cdots\!70 \nu^{3} - 96\!\cdots\!40 \nu^{2} - 20\!\cdots\!80 \nu + 21\!\cdots\!00\right) x - 14\!\cdots\!94 \nu^{5} - 69\!\cdots\!22 \nu^{4} + 39\!\cdots\!26 \nu^{3} + 10\!\cdots\!62 \nu^{2} + 22\!\cdots\!04 \nu - 23\!\cdots\!14}$
\(\displaystyle \phi(t,x) = \frac{1}{5^{5}}(483061 \nu^{5} + 391963 \nu^{4} - 1231879 \nu^{3} - 3745623 \nu^{2} - 8933366 \nu + 5352636) \, t\)
Embeddings
Each permutation triple in the orbit corresponds to an embedded Belyi map with coefficients in $\mathbb{C}$. The table below gives this correspondence.
| Embedding $\nu \mapsto \nu_i \in \mathbb{C}$ | Permutation triple | $-0.7150809416026352+1.862607205725975\sqrt{-1}$ | $(1,2,6,5,7,8), (1,3)(2,8,7)(4,6,5), (1,2,3)(4,5)(6,7)$ | $-2.256884530523702+0.0\sqrt{-1}$ | $(1,8,2,6,5,7), (1,3,8)(2,7)(4,6,5), (1,2,3)(4,5)(6,7)$ | $0.7430469970567619+0.0\sqrt{-1}$ | $(1,8,3,6,5,7), (1,8)(2,7,3)(4,6,5), (1,2,3)(4,5)(6,7)$ | $-0.7150809416026352-1.862607205725975\sqrt{-1}$ | $(1,2,7,4,5,3), (1,8)(2,3,7)(4,5,6), (1,2,8)(3,4)(5,6)$ | $0.4879476789894319+0.0\sqrt{-1}$ | $(1,4,3,7,2,8), (1,4,5)(2,8)(3,7,6), (1,2,3)(4,5)(6,7)$ | $2.456051737682778+0.0\sqrt{-1}$ | $(1,6,8,7,2,5), (1,3,7)(2,5,4)(6,8), (1,2,3)(4,5)(6,7)$ |