Passport invariants
| Degree: | $9$ |
| Monodromy group: | $S_9$ |
| 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 |
| $8$ | $8, 1$ |
| $6$ | $3, 2, 1, 1, 1, 1$ |
| $7$ | $7, 1, 1$ |
Base field
10.2.11014188167187136512.1 ; Generator \(\nu\), with minimal polynomial \( T^{10} - 4 T^{9} + 12 T^{8} - 16 T^{7} + 28 T^{6} - 140 T^{4} + 320 T^{3} - 300 T^{2} + 144 T - 24 \).
Curve
| $\mathbb{P}^1$, with affine coordinate $x$ | |
|
$\displaystyle \left(-1\right) x^{8} \left(1/28 \left(-25 \nu^{9} + 75 \nu^{8} - 211 \nu^{7} + 164 \nu^{6} - 446 \nu^{5} - 438 \nu^{4} + 3350 \nu^{3} - 4104 \nu^{2} + 2328 \nu - 420\right) x + 1/28 \left(-48 \nu^{9} + 97 \nu^{8} - 121 \nu^{7} - 656 \nu^{6} + 1520 \nu^{5} - 5618 \nu^{4} + 15674 \nu^{3} - 14864 \nu^{2} + 7064 \nu - 500\right)\right) t + \left(1/14 \left(4132455 \nu^{9} - 24039150 \nu^{8} + 93247913 \nu^{7} - 235402639 \nu^{6} + 542905117 \nu^{5} - 985526422 \nu^{4} + 1211396810 \nu^{3} - 878716086 \nu^{2} + 357336924 \nu - 54539878\right)\right) \left(1/28 \left(-9884 \nu^{9} + 42874 \nu^{8} - 122500 \nu^{7} + 168769 \nu^{6} - 246612 \nu^{5} + 27436 \nu^{4} + 1558584 \nu^{3} - 3454698 \nu^{2} + 2798696 \nu - 579412\right) x^{2} + 1/14 \left(-214 \nu^{9} + 815 \nu^{8} - 2349 \nu^{7} + 2788 \nu^{6} - 4931 \nu^{5} - 1308 \nu^{4} + 30830 \nu^{3} - 61294 \nu^{2} + 44550 \nu - 11526\right) x + 1/4 \left(79731 \nu^{9} - 294370 \nu^{8} + 866110 \nu^{7} - 1008926 \nu^{6} + 1921596 \nu^{5} + 592172 \nu^{4} - 10980892 \nu^{3} + 22133848 \nu^{2} - 17104868 \nu + 6214884\right)\right)=0$
|
Map
\(\displaystyle \phi(x) =\)
$\displaystyle 2401 \frac{\left(63\!\cdots\!72 \nu^{9} - 24\!\cdots\!96 \nu^{8} + 72\!\cdots\!80 \nu^{7} - 92\!\cdots\!68 \nu^{6} + 16\!\cdots\!12 \nu^{5} + 22\!\cdots\!20 \nu^{4} - 89\!\cdots\!80 \nu^{3} + 19\!\cdots\!96 \nu^{2} - 16\!\cdots\!04 \nu + 58\!\cdots\!76\right) x^{9} + \left(20\!\cdots\!05 \nu^{9} - 75\!\cdots\!36 \nu^{8} + 21\!\cdots\!40 \nu^{7} - 24\!\cdots\!93 \nu^{6} + 44\!\cdots\!53 \nu^{5} + 17\!\cdots\!00 \nu^{4} - 29\!\cdots\!60 \nu^{3} + 55\!\cdots\!26 \nu^{2} - 36\!\cdots\!10 \nu + 49\!\cdots\!24\right) x^{8}}{39\!\cdots\!76 x^{9} + \left(41\!\cdots\!05 \nu^{9} - 23\!\cdots\!72 \nu^{8} + 83\!\cdots\!00 \nu^{7} - 16\!\cdots\!97 \nu^{6} + 24\!\cdots\!53 \nu^{5} - 17\!\cdots\!20 \nu^{4} - 58\!\cdots\!96 \nu^{3} + 24\!\cdots\!70 \nu^{2} - 44\!\cdots\!30 \nu + 10\!\cdots\!56\right) x^{8} + \left(38\!\cdots\!76 \nu^{9} - 14\!\cdots\!92 \nu^{8} + 40\!\cdots\!76 \nu^{7} - 42\!\cdots\!28 \nu^{6} + 69\!\cdots\!88 \nu^{5} + 42\!\cdots\!12 \nu^{4} - 57\!\cdots\!00 \nu^{3} + 11\!\cdots\!20 \nu^{2} - 32\!\cdots\!52 \nu + 10\!\cdots\!40\right) x^{7} + \left(-30\!\cdots\!68 \nu^{9} + 16\!\cdots\!00 \nu^{8} - 46\!\cdots\!60 \nu^{7} + 70\!\cdots\!40 \nu^{6} - 88\!\cdots\!84 \nu^{5} + 28\!\cdots\!72 \nu^{4} + 49\!\cdots\!64 \nu^{3} - 15\!\cdots\!04 \nu^{2} + 10\!\cdots\!76 \nu - 18\!\cdots\!24\right) x^{6} + \left(-20\!\cdots\!08 \nu^{9} + 32\!\cdots\!80 \nu^{8} - 62\!\cdots\!28 \nu^{7} - 12\!\cdots\!14 \nu^{6} - 86\!\cdots\!76 \nu^{5} - 69\!\cdots\!32 \nu^{4} + 27\!\cdots\!28 \nu^{3} + 19\!\cdots\!12 \nu^{2} - 39\!\cdots\!96 \nu + 96\!\cdots\!28\right) x^{5} + \left(21\!\cdots\!76 \nu^{9} - 73\!\cdots\!72 \nu^{8} + 18\!\cdots\!68 \nu^{7} - 16\!\cdots\!66 \nu^{6} + 31\!\cdots\!34 \nu^{5} + 23\!\cdots\!96 \nu^{4} - 33\!\cdots\!92 \nu^{3} + 38\!\cdots\!80 \nu^{2} - 11\!\cdots\!60 \nu + 98\!\cdots\!00\right) x^{4} + \left(-37\!\cdots\!20 \nu^{9} + 16\!\cdots\!60 \nu^{8} - 42\!\cdots\!84 \nu^{7} + 58\!\cdots\!00 \nu^{6} - 65\!\cdots\!56 \nu^{5} + 26\!\cdots\!28 \nu^{4} + 69\!\cdots\!44 \nu^{3} - 11\!\cdots\!20 \nu^{2} + 62\!\cdots\!52 \nu - 10\!\cdots\!00\right) x^{3} + \left(-16\!\cdots\!60 \nu^{9} + 28\!\cdots\!64 \nu^{8} - 79\!\cdots\!72 \nu^{7} - 39\!\cdots\!48 \nu^{6} - 19\!\cdots\!00 \nu^{5} - 54\!\cdots\!12 \nu^{4} + 18\!\cdots\!80 \nu^{3} + 36\!\cdots\!36 \nu^{2} - 88\!\cdots\!96 \nu + 18\!\cdots\!36\right) x^{2} + \left(-75\!\cdots\!64 \nu^{9} + 15\!\cdots\!60 \nu^{8} - 49\!\cdots\!44 \nu^{7} + 32\!\cdots\!79 \nu^{6} - 13\!\cdots\!24 \nu^{5} - 26\!\cdots\!80 \nu^{4} + 72\!\cdots\!72 \nu^{3} - 60\!\cdots\!82 \nu^{2} + 13\!\cdots\!36 \nu - 33\!\cdots\!96\right) x + 45\!\cdots\!81 \nu^{9} - 34\!\cdots\!60 \nu^{8} + 75\!\cdots\!88 \nu^{7} - 16\!\cdots\!97 \nu^{6} + 62\!\cdots\!53 \nu^{5} - 32\!\cdots\!12 \nu^{4} - 14\!\cdots\!96 \nu^{3} + 24\!\cdots\!98 \nu^{2} - 17\!\cdots\!46 \nu + 34\!\cdots\!20}$
\(\displaystyle \phi(t,x) = \frac{1}{2^{2} \cdot 3^{6} \cdot 7^{4}}(-16955278642122514198028 \nu^{9} + 62600513351005584629490 \nu^{8} - 184188383950321976766420 \nu^{7} + 214572085293247858879703 \nu^{6} - 408680162752594617570840 \nu^{5} - 125834323410946793749344 \nu^{4} + 2334994076553322634454760 \nu^{3} - 4706734689137057496269310 \nu^{2} + 3637360256208305636575824 \nu - 1321601663574684715201248) \, t\)
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 | $1.700929490067345-2.253943635463972\sqrt{-1}$ | $(1,7,6,5,4,2,8,9), (2,9,8)(3,4), (1,2,3,4,5,6,7)$ | $1.700929490067345+2.253943635463972\sqrt{-1}$ | $(1,6,5,4,3,2,7,8), (1,9)(2,8,7), (1,2,3,4,5,6,9)$ | $-0.6290335698006824-2.164723157298074\sqrt{-1}$ | $(1,7,6,5,3,2,8,9), (2,9,8)(4,5), (1,2,3,4,5,6,7)$ | $-1.815901262051668+0.0\sqrt{-1}$ | $(1,7,6,4,3,2,8,9), (2,9,8)(5,6), (1,2,3,4,5,6,7)$ | $-0.6290335698006824+2.164723157298074\sqrt{-1}$ | $(1,7,5,4,3,2,8,9), (2,9,8)(6,7), (1,2,3,4,5,6,7)$ | $1.134170581191287+0.7201189121558191\sqrt{-1}$ | $(1,9,7,6,5,4,3,8), (1,9)(2,8,3), (1,2,3,4,5,6,7)$ | $1.134170581191287-0.7201189121558191\sqrt{-1}$ | $(1,8,6,5,4,3,2,7), (1,9,8)(2,7), (1,2,3,4,5,6,9)$ | $0.5479320358256109+0.5354516496387348\sqrt{-1}$ | $(1,9,7,6,5,4,8,2), (1,9)(3,8,4), (1,2,3,4,5,6,7)$ | $0.5479320358256109-0.5354516496387348\sqrt{-1}$ | $(1,8,6,5,4,3,7,2), (1,9,8)(3,7), (1,2,3,4,5,6,9)$ | $0.3079041874845466+0.0\sqrt{-1}$ | $(1,7,9,6,5,4,8,2), (3,8,4)(7,9), (1,2,3,4,5,6,7)$ |