Invariants
Base field: | $\F_{2}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - x + 2 x^{2} )( 1 - x - x^{2} - 2 x^{3} + 4 x^{4} )$ |
$1 - 2 x + 2 x^{2} - 3 x^{3} + 4 x^{4} - 8 x^{5} + 8 x^{6}$ | |
Frobenius angles: | $\pm0.0516399385854$, $\pm0.384973271919$, $\pm0.718306605252$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $1$ |
Isomorphism classes: | 2 |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2$ | $56$ | $224$ | $4144$ | $16522$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $1$ | $5$ | $4$ | $17$ | $11$ | $38$ | $155$ | $257$ | $508$ | $1025$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2+(x^4+x^2+x)y=x^8+x^6+x^4+x^3+x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{6}}$.
Endomorphism algebra over $\F_{2}$The isogeny class factors as 1.2.ab $\times$ 2.2.ab_ab 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.aj 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$ |
- Endomorphism algebra over $\F_{2^{2}}$
The base change of $A$ to $\F_{2^{2}}$ is 1.4.d $\times$ 2.4.ad_f. The endomorphism algebra for each factor is: - Endomorphism algebra over $\F_{2^{3}}$
The base change of $A$ to $\F_{2^{3}}$ is 1.8.af 2 $\times$ 1.8.f. The endomorphism algebra for each factor is: - 1.8.af 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
- 1.8.f : \(\Q(\sqrt{-7}) \).
Base change
This is a primitive isogeny class.