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 |
| $6$ | $6, 1, 1$ |
| $4$ | $4, 2, 2$ |
| $6$ | $3, 2, 2, 1$ |
Base field
6.2.11430720.4 ; Generator \(\nu\), with minimal polynomial \( T^{6} - 3 T^{5} - 3 T^{4} + 4 T^{3} - 6 T + 2 \).
Curve
| $\mathbb{P}^1$, with affine coordinate $x$ | |
|
$\displaystyle \left(1/7 \left(-\nu^{5} + 6 \nu^{4} - \nu^{3} - 8 \nu^{2} + 10 \nu - 3\right)\right) x^{4} \left(1/7 \left(6 \nu^{5} - 15 \nu^{4} - 22 \nu^{3} - \nu^{2} + 3 \nu - 31\right) x^{2} + 1/7 \left(-80 \nu^{5} + 214 \nu^{4} + 312 \nu^{3} - 220 \nu^{2} - 96 \nu + 432\right) x + 1/7 \left(201 \nu^{5} - 534 \nu^{4} - 786 \nu^{3} + 537 \nu^{2} + 174 \nu - 1161\right)\right)^{2} t + \left(1/7 \left(-13 \nu^{5} + 36 \nu^{4} + 50 \nu^{3} - 34 \nu^{2} - 10 \nu + 73\right)\right) \left(1/7 \left(-65 \nu^{5} + 208 \nu^{4} + 194 \nu^{3} - 352 \nu^{2} - 190 \nu + 120\right) x^{2} + 1/7 \left(-100 \nu^{5} + 236 \nu^{4} + 488 \nu^{3} - 128 \nu^{2} - 344 \nu + 120\right) x + 1/7 \left(-4 \nu^{5} + 17 \nu^{4} + 10 \nu^{3} - 60 \nu^{2} - 44 \nu + 23\right)\right)=0$
|
Map
\(\displaystyle \phi(x) =\)
$\displaystyle \frac{20}{87779} \frac{\left(63\!\cdots\!28 \nu^{5} - 15\!\cdots\!92 \nu^{4} - 30\!\cdots\!92 \nu^{3} + 16\!\cdots\!84 \nu^{2} + 30\!\cdots\!36 \nu - 10\!\cdots\!09\right) x^{8} + \left(41\!\cdots\!60 \nu^{5} - 21\!\cdots\!40 \nu^{4} - 31\!\cdots\!60 \nu^{3} + 84\!\cdots\!00 \nu^{2} + 61\!\cdots\!40 \nu - 40\!\cdots\!00\right) x^{7} + \left(-38\!\cdots\!85 \nu^{5} + 12\!\cdots\!30 \nu^{4} + 13\!\cdots\!20 \nu^{3} - 30\!\cdots\!10 \nu^{2} - 29\!\cdots\!10 \nu + 13\!\cdots\!80\right) x^{6}}{94\!\cdots\!00 x^{8} + \left(66\!\cdots\!20 \nu^{5} - 17\!\cdots\!80 \nu^{4} - 31\!\cdots\!60 \nu^{3} + 20\!\cdots\!20 \nu^{2} + 33\!\cdots\!60 \nu - 14\!\cdots\!20\right) x^{7} + \left(63\!\cdots\!08 \nu^{5} - 19\!\cdots\!12 \nu^{4} - 24\!\cdots\!84 \nu^{3} + 38\!\cdots\!48 \nu^{2} + 42\!\cdots\!84 \nu - 18\!\cdots\!68\right) x^{6} + \left(-24\!\cdots\!72 \nu^{5} + 52\!\cdots\!08 \nu^{4} + 13\!\cdots\!56 \nu^{3} - 16\!\cdots\!32 \nu^{2} - 92\!\cdots\!56 \nu + 46\!\cdots\!12\right) x^{5} + \left(-52\!\cdots\!30 \nu^{5} + 31\!\cdots\!45 \nu^{4} - 39\!\cdots\!10 \nu^{3} - 12\!\cdots\!30 \nu^{2} - 82\!\cdots\!40 \nu + 45\!\cdots\!30\right) x^{4} + \left(40\!\cdots\!60 \nu^{5} - 80\!\cdots\!40 \nu^{4} - 23\!\cdots\!80 \nu^{3} - 13\!\cdots\!40 \nu^{2} + 14\!\cdots\!80 \nu - 84\!\cdots\!60\right) x^{3} + \left(-94\!\cdots\!60 \nu^{5} + 29\!\cdots\!90 \nu^{4} + 78\!\cdots\!80 \nu^{3} + 10\!\cdots\!40 \nu^{2} + 18\!\cdots\!20 \nu - 21\!\cdots\!40\right) x^{2} + \left(-28\!\cdots\!52 \nu^{5} + 59\!\cdots\!28 \nu^{4} + 15\!\cdots\!96 \nu^{3} - 79\!\cdots\!12 \nu^{2} - 10\!\cdots\!96 \nu + 63\!\cdots\!92\right) x + 77\!\cdots\!38 \nu^{5} - 13\!\cdots\!07 \nu^{4} - 47\!\cdots\!74 \nu^{3} - 17\!\cdots\!22 \nu^{2} + 19\!\cdots\!24 \nu - 42\!\cdots\!98}$
\(\displaystyle \phi(t,x) = \frac{1}{2^{2} \cdot 5^{3} \cdot 7^{3}}(-874061 \nu^{5} + 2313952 \nu^{4} + 3438255 \nu^{3} - 2283819 \nu^{2} - 805854 \nu + 4959597) \, t\)
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 | $-1.159675608581792+0.4591215284725237\sqrt{-1}$ | $(1,2,5,6,8,7), (1,3)(2,7,5,4)(6,8), (1,2,3)(4,5)(6,7)$ | $0.7030484091981257-0.7281955863737308\sqrt{-1}$ | $(1,2,6,5,8,7), (1,3)(2,7)(4,6,8,5), (1,2,3)(4,5)(6,7)$ | $0.3523994070369582+0.0\sqrt{-1}$ | $(1,3,4,6,5,2), (1,8,7,2)(3,5)(4,6), (1,7,8)(2,3)(4,5)$ | $-1.159675608581792-0.4591215284725237\sqrt{-1}$ | $(1,2,3,7,4,5), (1,8)(2,5,6,4)(3,7), (1,2,8)(3,4)(5,6)$ | $0.7030484091981257+0.7281955863737308\sqrt{-1}$ | $(1,2,4,7,5,3), (1,8)(2,3)(4,5,6,7), (1,2,8)(3,4)(5,6)$ | $3.560854991730375+0.0\sqrt{-1}$ | $(1,4,2,6,8,7), (1,3,4,5)(2,7)(6,8), (1,2,3)(4,5)(6,7)$ |