Invariants
| Base field: | $\F_{2^{3}}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - x + 8 x^{2} )( 1 - 9 x + 35 x^{2} - 72 x^{3} + 64 x^{4} )$ |
| $1 - 10 x + 52 x^{2} - 179 x^{3} + 416 x^{4} - 640 x^{5} + 512 x^{6}$ | |
| Frobenius angles: | $\pm0.0373126015494$, $\pm0.296020731784$, $\pm0.443432958871$ |
| Angle rank: | $2$ (numerical) |
| Isomorphism classes: | 15 |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $3$ |
| Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $152$ | $275120$ | $140091104$ | $66977964000$ | $34588608746632$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-1$ | $69$ | $536$ | $3993$ | $32209$ | $261078$ | $2095099$ | $16767857$ | $134194568$ | $1073767989$ |
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 $\F_{2^{3}}$| The isogeny class factors as 1.8.ab $\times$ 2.8.aj_bj and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
The base change of $A$ to $\F_{2^{18}}$ is 1.262144.abeb 2 $\times$ 1.262144.tb. The endomorphism algebra for each factor is:
|
- Endomorphism algebra over $\F_{2^{6}}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.p $\times$ 2.64.al_cf. The endomorphism algebra for each factor is: - Endomorphism algebra over $\F_{2^{9}}$
The base change of $A$ to $\F_{2^{9}}$ is 1.512.x $\times$ 2.512.a_abeb. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.