Invariants
| Base field: | $\F_{2^{3}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 3 x + 8 x^{2} )( 1 + 8 x^{2} )$ |
| $1 - 3 x + 16 x^{2} - 24 x^{3} + 64 x^{4}$ | |
| Frobenius angles: | $\pm0.322067999368$, $\pm0.5$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $3$ |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
| $p$-rank: | $1$ |
| Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $54$ | $5832$ | $286254$ | $16574544$ | $1069776774$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $88$ | $558$ | $4048$ | $32646$ | $262168$ | $2095134$ | $16770976$ | $134239734$ | $1073857768$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+xy=(a+1)x^5+(a+1)x^3+x^2+x$
- $y^2+xy=(a^2+a+1)x^5+(a^2+a+1)x^3+x^2+x$
- $y^2+xy=(a^2+1)x^5+(a^2+1)x^3+x^2+x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{6}}$.
Endomorphism algebra over $\F_{2^{3}}$| The isogeny class factors as 1.8.ad $\times$ 1.8.a 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 $\F_{2^{6}}$ is 1.64.h $\times$ 1.64.q. The endomorphism algebra for each factor is:
|
Base change
This is a primitive isogeny class.