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 |
| $6$ | $6, 2$ |
| $6$ | $3, 3, 2$ |
| $6$ | $3, 2, 1, 1, 1$ |
Base field
6.2.7464960.4 ; Generator \(\nu\), with minimal polynomial \( T^{6} - 3 T^{4} - 8 T^{3} - 3 T^{2} + 5 \).
Curve
| $\mathbb{P}^1$, with affine coordinate $x$ | |
|
$\displaystyle \left(1/2 \left(\nu^{5} - 2 \nu^{4} - 2 \nu^{3} + 3 \nu + 6\right) x + 1/2 \left(-\nu^{4} + 2 \nu^{2} + 4 \nu - 1\right)\right)^{2} t + \left(1/4 \left(-15 \nu^{5} + 33 \nu^{4} + 6 \nu^{3} + 30 \nu^{2} - 29 \nu - 1\right)\right) x^{3} \left(x + 1/4 \left(\nu^{5} - \nu^{4} - 2 \nu^{3} - 6 \nu^{2} - 5 \nu - 15\right)\right)^{2} \left(x^{3} + \left(\nu^{5} + 2 \nu^{4} - 4 \nu^{3} - 14 \nu^{2} - 15 \nu - 2\right) x^{2} + 1/4 \left(-5 \nu^{5} - 23 \nu^{4} - 10 \nu^{3} + 114 \nu^{2} + 245 \nu + 187\right) x + 1/2 \left(47 \nu^{5} - 23 \nu^{4} - 90 \nu^{3} - 102 \nu^{2} + 225 \nu + 7\right)\right)=0$
|
Map
\(\displaystyle \phi(x) =\)
$\displaystyle 29\!\cdots\!25 \frac{\left(-89\!\cdots\!82 \nu^{5} - 23\!\cdots\!40 \nu^{4} + 21\!\cdots\!76 \nu^{3} + 72\!\cdots\!28 \nu^{2} + 59\!\cdots\!30 \nu + 57\!\cdots\!44\right) x^{8} + \left(-36\!\cdots\!10 \nu^{5} - 26\!\cdots\!38 \nu^{4} + 63\!\cdots\!56 \nu^{3} + 23\!\cdots\!12 \nu^{2} + 17\!\cdots\!90 \nu + 10\!\cdots\!62\right) x^{7} + \left(-20\!\cdots\!37 \nu^{5} - 49\!\cdots\!59 \nu^{4} - 61\!\cdots\!22 \nu^{3} - 14\!\cdots\!70 \nu^{2} + 29\!\cdots\!05 \nu + 40\!\cdots\!19\right) x^{6}}{35\!\cdots\!00 x^{8} + \left(-44\!\cdots\!00 \nu^{5} + 85\!\cdots\!00 \nu^{4} - 78\!\cdots\!00 \nu^{3} - 81\!\cdots\!00 \nu^{2} - 15\!\cdots\!00 \nu - 98\!\cdots\!00\right) x^{7} + \left(-10\!\cdots\!60 \nu^{5} + 16\!\cdots\!30 \nu^{4} + 11\!\cdots\!40 \nu^{3} + 24\!\cdots\!60 \nu^{2} + 20\!\cdots\!00 \nu + 97\!\cdots\!50\right) x^{6} + \left(67\!\cdots\!42 \nu^{5} + 18\!\cdots\!54 \nu^{4} + 27\!\cdots\!72 \nu^{3} + 12\!\cdots\!28 \nu^{2} + 16\!\cdots\!10 \nu - 10\!\cdots\!30\right) x^{5} + \left(-31\!\cdots\!95 \nu^{5} - 79\!\cdots\!15 \nu^{4} - 10\!\cdots\!70 \nu^{3} - 27\!\cdots\!30 \nu^{2} + 23\!\cdots\!75 \nu + 61\!\cdots\!75\right) x^{4} + \left(-13\!\cdots\!02 \nu^{5} - 34\!\cdots\!10 \nu^{4} - 47\!\cdots\!44 \nu^{3} - 11\!\cdots\!64 \nu^{2} + 10\!\cdots\!66 \nu + 26\!\cdots\!70\right) x^{3} + \left(-11\!\cdots\!81 \nu^{5} - 28\!\cdots\!05 \nu^{4} - 38\!\cdots\!82 \nu^{3} - 97\!\cdots\!42 \nu^{2} + 86\!\cdots\!73 \nu + 21\!\cdots\!85\right) x^{2} + \left(23\!\cdots\!34 \nu^{5} + 58\!\cdots\!50 \nu^{4} + 80\!\cdots\!08 \nu^{3} + 20\!\cdots\!68 \nu^{2} - 17\!\cdots\!02 \nu - 45\!\cdots\!90\right) x + 64\!\cdots\!49 \nu^{5} + 16\!\cdots\!27 \nu^{4} + 22\!\cdots\!22 \nu^{3} + 56\!\cdots\!70 \nu^{2} - 49\!\cdots\!29 \nu - 12\!\cdots\!95}$
\(\displaystyle \phi(t,x) = \frac{1}{2^{2} \cdot 5^{5}}(-500643 \nu^{5} - 1370091 \nu^{4} + 5810762 \nu^{3} + 1635138 \nu^{2} + 3883635 \nu - 5216805) \, t\)
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 | $2.546597033590813+0.0\sqrt{-1}$ | $(1,8,3,6,5,7)(2,4), (1,8)(2,7,5)(3,4,6), (1,2,3)(4,5)$ | $-0.4943072214862377+0.9529561870660095\sqrt{-1}$ | $(1,5,6,2,7,8)(3,4), (1,4)(2,8,7)(3,6,5), (1,2,3)(4,5)$ | $-1.133739784131182-1.056338600666304\sqrt{-1}$ | $(1,2,3,5,4,6)(7,8), (1,7,8)(2,6,4)(3,5), (1,2,8)(3,4)$ | $-0.4943072214862377-0.9529561870660095\sqrt{-1}$ | $(1,6,5,2,7,8)(3,4), (1,4,6)(2,8,7)(3,5), (1,2,3)(4,5)$ | $0.709496977644027+0.0\sqrt{-1}$ | $(1,2,4,5,6,3)(7,8), (1,7,8)(2,3)(4,6,5), (1,2,8)(3,4)$ | $-1.133739784131182+1.056338600666304\sqrt{-1}$ | $(1,2,5,4,6,3)(7,8), (1,7,8)(2,3,5)(4,6), (1,2,8)(3,4)$ |