Invariants
Base field: | $\F_{2}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 2 x + 2 x^{2} )( 1 + x + 2 x^{2} )$ |
$1 - x + 2 x^{2} - 2 x^{3} + 4 x^{4}$ | |
Frobenius angles: | $\pm0.250000000000$, $\pm0.615026728081$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $1$ |
Isomorphism classes: | 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $4$ | $40$ | $52$ | $400$ | $1804$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $2$ | $8$ | $8$ | $24$ | $52$ | $56$ | $100$ | $256$ | $476$ | $968$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2+xy=x^5+x^3+x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{4}}$.
Endomorphism algebra over $\F_{2}$The isogeny class factors as 1.2.ac $\times$ 1.2.b 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 $\F_{2^{4}}$ is 1.16.ab $\times$ 1.16.i. 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. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.