Passport invariants
| Degree: | $7$ |
| Monodromy group: | $A_7$ |
| 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 |
| $5$ | $5, 1, 1$ |
| $3$ | $3, 3, 1$ |
| $4$ | $4, 2, 1$ |
Base field
5.1.1512000.1 ; Generator \(\nu\), with minimal polynomial \( T^{5} - T^{4} + 2 T^{3} + 2 T^{2} - 20 T - 20 \).
Curve
| $\mathbb{P}^1$, with affine coordinate $x$ | |
|
$\displaystyle \left(-1\right) \left(\left(-2 \nu^{4} + 4 \nu^{3} - 13 \nu^{2} + 4 \nu + 40\right) x^{2} + 1/6 \left(\nu^{4} + 3 \nu^{3} - 4 \nu^{2} + 10 \nu - 58\right) x + 1/2 \left(\nu^{4} - 2 \nu^{3} + 5 \nu^{2} - 4 \nu - 14\right)\right) t + \left(1/3 \left(-\nu^{4} + 3 \nu^{3} + 4 \nu^{2} - 10 \nu - 11\right)\right) x^{4} \left(x + 1/2 \left(2 \nu^{4} - 5 \nu^{3} + 11 \nu^{2} - 12 \nu - 26\right)\right)^{2} \left(1/6 \left(-\nu^{4} + 3 \nu^{3} - 8 \nu^{2} + 8 \nu + 10\right) x + 1/2 \left(-8 \nu^{4} + 17 \nu^{3} - 35 \nu^{2} + 22 \nu + 138\right)\right)=0$
|
Map
\(\displaystyle \phi(x) =\)
$\displaystyle \frac{1728}{51341} \frac{\left(98\!\cdots\!93 \nu^{4} + 40\!\cdots\!44 \nu^{3} - 33\!\cdots\!86 \nu^{2} - 18\!\cdots\!40 \nu - 12\!\cdots\!04\right) x^{7} + \left(14\!\cdots\!36 \nu^{4} + 54\!\cdots\!80 \nu^{3} - 56\!\cdots\!72 \nu^{2} - 20\!\cdots\!08 \nu - 10\!\cdots\!60\right) x^{6} + \left(12\!\cdots\!08 \nu^{4} - 12\!\cdots\!48 \nu^{3} - 41\!\cdots\!16 \nu^{2} + 48\!\cdots\!72 \nu + 42\!\cdots\!44\right) x^{5}}{29\!\cdots\!69 x^{7} + \left(60\!\cdots\!34 \nu^{4} + 25\!\cdots\!28 \nu^{3} - 55\!\cdots\!60 \nu^{2} + 21\!\cdots\!88 \nu + 12\!\cdots\!35\right) x^{6} + \left(21\!\cdots\!29 \nu^{4} + 21\!\cdots\!40 \nu^{3} + 49\!\cdots\!70 \nu^{2} - 64\!\cdots\!72 \nu - 52\!\cdots\!39\right) x^{5} + \left(-22\!\cdots\!85 \nu^{4} + 47\!\cdots\!00 \nu^{3} + 98\!\cdots\!70 \nu^{2} - 96\!\cdots\!40 \nu - 13\!\cdots\!25\right) x^{4} + \left(-10\!\cdots\!05 \nu^{4} - 48\!\cdots\!60 \nu^{3} - 18\!\cdots\!30 \nu^{2} + 14\!\cdots\!60 \nu + 12\!\cdots\!75\right) x^{3} + \left(45\!\cdots\!09 \nu^{4} + 86\!\cdots\!52 \nu^{3} - 21\!\cdots\!38 \nu^{2} - 58\!\cdots\!60 \nu - 34\!\cdots\!79\right) x^{2} + \left(28\!\cdots\!71 \nu^{4} + 37\!\cdots\!56 \nu^{3} - 83\!\cdots\!86 \nu^{2} - 16\!\cdots\!48 \nu - 65\!\cdots\!45\right) x - 42\!\cdots\!63 \nu^{4} - 12\!\cdots\!44 \nu^{3} + 42\!\cdots\!82 \nu^{2} + 60\!\cdots\!52 \nu + 47\!\cdots\!29}$
\(\displaystyle \phi(t,x) = \frac{1}{2^{2} \cdot 3^{3}}(-365 \nu^{4} - 1065 \nu^{3} + 1008 \nu^{2} + 4475 \nu + 2895) \, t\)
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 | $-1.194505016570467+0.1926707022709436\sqrt{-1}$ | $(1,2,5,6,7), (1,4,3)(2,7,6), (1,2,3,4)(5,6)$ | $-1.194505016570467-0.1926707022709436\sqrt{-1}$ | $(1,2,7,5,6), (1,4,3)(2,6,7), (1,2,3,4)(5,6)$ | $2.250564802522955+0.0\sqrt{-1}$ | $(1,7,4,5,6), (1,3,7)(2,6,4), (1,2,4,3)(5,6)$ | $0.5692226153089897+2.397134870226378\sqrt{-1}$ | $(1,7,4,5,2), (1,3,7)(4,5,6), (1,2,4,3)(5,6)$ | $0.5692226153089897-2.397134870226378\sqrt{-1}$ | $(1,7,4,2,5), (1,3,7)(2,5,6), (1,2,4,3)(5,6)$ |