This Belyi map is the lowest degree map of hyperbolic geometry type.
Passport invariants
| Degree: | $4$ |
| Monodromy group: | $S_4$ |
| Genus: | $1$ |
| 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 |
| $4$ | $4$ |
| $4$ | $4$ |
| $3$ | $3, 1$ |
Base field
Curve
|
$\displaystyle y^{2} = x^{3} + \frac{47}{768} x + \frac{2359}{55296}$
, isomorphic to elliptic curve with label 48.a6
|
|
|
$\displaystyle t^{2} + \left(2 x^{2} - 4 x + 9\right) t + x^{4}=0$
, isomorphic to elliptic curve with label 48.a6
|
(smooth)
(planar)
Map
\(\displaystyle \phi(x,y) =\)
$\displaystyle 972 \frac{-2304 x^{2} + 96 x + 2304 y + 467}{5308416 x^{4} - 5750784 x^{3} - 2142720 x^{2} - 235200 x - 8375}$
\(\displaystyle \phi(t,x) = -\frac{2^{2}}{3^{3}} \, t\)
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 | Dessin |
| $\text{not applicable (over $\mathbb{Q}$)}$ | $(1,2,3,4), (1,3,4,2), (1,3,4)$ | View Embedding |