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 |
| $7$ | $7$ |
| $2$ | $2, 2, 1, 1, 1$ |
| $5$ | $5, 1, 1$ |
Base field
\(\Q(\sqrt{21}) \) ; Generator \(\nu\), with minimal polynomial \( T^{2} - T - 5 \).
Curve
| $\mathbb{P}^1$, with affine coordinate $x$ | |
|
$\displaystyle x^{7} t + \left(\nu + 2\right) \left(\left(-3 \nu + 15\right) x^{2} + \left(3 \nu - 5\right) x - 4 \nu + 12\right)=0$
|
Map
\(\displaystyle \phi(x) =\)
$\displaystyle 5184 \frac{\left(-224 \nu + 619\right) x^{7}}{102 x^{7} + \left(28 \nu - 2485\right) x^{6} + \left(-672 \nu + 23940\right) x^{5} + \left(6300 \nu - 112875\right) x^{4} + \left(-28000 \nu + 253750\right) x^{3} + \left(52500 \nu - 196875\right) x^{2} - 87500 \nu - 203125}$
\(\displaystyle \phi(t,x) = \frac{2^{6} \cdot 7^{3}}{3^{5} \cdot 5^{6}}(64 \nu - 179) \, t\)
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 | $2.79128784747792+0.0\sqrt{-1}$ | $(1,2,7,6,5,4,3), (1,2)(3,4), (1,4,5,6,7)$ | $-1.79128784747792+0.0\sqrt{-1}$ | $(1,2,7,6,4,3,5), (1,2)(3,4), (1,5,4,6,7)$ |