Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $6$ |
| L-polynomial: | $1 - 8 x^{6} + 64 x^{12}$ |
| Frobenius angles: | $\pm0.0555555555556$, $\pm0.277777777778$, $\pm0.388888888889$, $\pm0.611111111111$, $\pm0.722222222222$, $\pm0.944444444444$ |
| Angle rank: | $0$ (numerical) |
| Number field: | 12.0.101559956668416.1 |
| Galois group: | $C_6\times C_2$ |
| Jacobians: | $0$ |
| Cyclic group of points: | yes |
This isogeny class is simple but not geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable.
Newton polygon
This isogeny class is supersingular.
| $p$-rank: | $0$ |
| Slopes: | $[1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $57$ | $3249$ | $185193$ | $17313921$ | $1073709057$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $3$ | $5$ | $9$ | $17$ | $33$ | $17$ | $129$ | $257$ | $513$ | $1025$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{18}}$.
Endomorphism algebra over $\F_{2}$| The endomorphism algebra of this simple isogeny class is 12.0.101559956668416.1. |
| The base change of $A$ to $\F_{2^{18}}$ is 1.262144.bnk 6 and its endomorphism algebra is $\mathrm{M}_{6}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$. |
- Endomorphism algebra over $\F_{2^{2}}$
The base change of $A$ to $\F_{2^{2}}$ is 3.4.a_a_ai 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\zeta_{9})\)$)$ - Endomorphism algebra over $\F_{2^{3}}$
The base change of $A$ to $\F_{2^{3}}$ is 2.8.a_ai 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-2}, \sqrt{-3})\)$)$ - Endomorphism algebra over $\F_{2^{6}}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.ai 6 and its endomorphism algebra is $\mathrm{M}_{6}($\(\Q(\sqrt{-3}) \)$)$ - Endomorphism algebra over $\F_{2^{9}}$
The base change of $A$ to $\F_{2^{9}}$ is 1.512.a 6 and its endomorphism algebra is $\mathrm{M}_{6}($\(\Q(\sqrt{-2}) \)$)$
Base change
This is a primitive isogeny class.