Invariants
Base field: | $\F_{2}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 2 x + 2 x^{2} )( 1 - x + 2 x^{2} )( 1 + 2 x^{2} )$ |
$1 - 3 x + 8 x^{2} - 12 x^{3} + 16 x^{4} - 12 x^{5} + 8 x^{6}$ | |
Frobenius angles: | $\pm0.250000000000$, $\pm0.384973271919$, $\pm0.5$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $0$ |
Isomorphism classes: | 3 |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $6$ | $360$ | $1638$ | $3600$ | $29766$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $0$ | $12$ | $18$ | $16$ | $30$ | $72$ | $126$ | $224$ | $486$ | $1032$ |
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 $\F_{2^{8}}$.
Endomorphism algebra over $\F_{2}$The isogeny class factors as 1.2.ac $\times$ 1.2.ab $\times$ 1.2.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 $\F_{2^{8}}$ is 1.256.abg 2 $\times$ 1.256.bf. The endomorphism algebra for each factor is:
|
- Endomorphism algebra over $\F_{2^{2}}$
The base change of $A$ to $\F_{2^{2}}$ is 1.4.a $\times$ 1.4.d $\times$ 1.4.e. The endomorphism algebra for each factor is: - 1.4.a : \(\Q(\sqrt{-1}) \).
- 1.4.d : \(\Q(\sqrt{-7}) \).
- 1.4.e : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- Endomorphism algebra over $\F_{2^{4}}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.ai $\times$ 1.16.ab $\times$ 1.16.i. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.