Invariants
Base field: | $\F_{2}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - x + 2 x^{2} )( 1 - 2 x^{2} )^{2}$ |
$1 - x - 2 x^{2} + 4 x^{3} - 4 x^{4} - 4 x^{5} + 8 x^{6}$ | |
Frobenius angles: | $0$, $0$, $\pm0.384973271919$, $1$, $1$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $0$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2$ | $8$ | $686$ | $1296$ | $21142$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $2$ | $0$ | $14$ | $0$ | $22$ | $24$ | $142$ | $224$ | $518$ | $840$ |
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 $\F_{2^{2}}$.
Endomorphism algebra over $\F_{2}$The isogeny class factors as 1.2.ab $\times$ 2.2.a_ae 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 $\F_{2^{2}}$ is 1.4.ae 2 $\times$ 1.4.d. The endomorphism algebra for each factor is:
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Base change
This is a primitive isogeny class.