Passport invariants
| Degree: | $3$ |
| Monodromy group: | $C_3$ |
| Genus: | $1$ |
| Geometry type: | Euclidean |
| 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 |
| $3$ | $3$ |
| $3$ | $3$ |
| $3$ | $3$ |
Base field
\(\Q(\sqrt{-3}) \) ; Generator \(\nu\), with minimal polynomial \( T^{2} - T + 1 \).
Curve
|
$\displaystyle y^{2} = x^{3} + 1$
, isomorphic to elliptic curve with label 2.0.3.1-144.1-CMa1
|
|
|
$\displaystyle t^{2} - t + 2 x^{3}=0$
, isomorphic to elliptic curve with label 2.0.3.1-144.1-CMa1
|
(smooth)
(planar)
Map
\(\displaystyle \phi(x,y) =\)
$\displaystyle \frac{1}{2} ( -y + 1 )$
\(\displaystyle \phi(t,x) = \, t\)
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 |
| $0.5+0.8660254037844387\sqrt{-1}$ | $(1,2,3), (1,2,3), (1,2,3)$ | View Embedding |