Invariants
| Base field: | $\F_{2^{3}}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - 4 x + 8 x^{2} )^{3}$ |
| $1 - 12 x + 72 x^{2} - 256 x^{3} + 576 x^{4} - 768 x^{5} + 512 x^{6}$ | |
| Frobenius angles: | $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$ |
| Angle rank: | $0$ (numerical) |
This isogeny class is not simple, not primitive, not ordinary, and supersingular. It is principally polarizable.
Newton polygon
This isogeny class is supersingular.
| $p$-rank: | $0$ |
| Slopes: | $[1/2, 1/2, 1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $125$ | $274625$ | $161878625$ | $75418890625$ | $36018736890625$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-3$ | $65$ | $609$ | $4481$ | $33537$ | $262145$ | $2091009$ | $16752641$ | $134168577$ | $1073741825$ |
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 $\F_{2^{12}}$.
Endomorphism algebra over $\F_{2^{3}}$| The isogeny class factors as 1.8.ae 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$ |
| The base change of $A$ to $\F_{2^{12}}$ is 1.4096.ey 3 and its endomorphism algebra is $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$. |
- Endomorphism algebra over $\F_{2^{6}}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.a 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{3}}$.
| Subfield | Primitive Model |
| $\F_{2}$ | 3.2.a_a_ae |
| $\F_{2}$ | 3.2.g_s_bg |