Invariants
Base field: | $\F_{2}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 + 2 x^{2} )( 1 + x + 2 x^{2} )$ |
$1 + x + 4 x^{2} + 2 x^{3} + 4 x^{4}$ | |
Frobenius angles: | $\pm0.5$, $\pm0.615026728081$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $0$ |
Isomorphism classes: | 1 |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $12$ | $72$ | $36$ | $144$ | $1452$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $12$ | $4$ | $8$ | $44$ | $72$ | $116$ | $256$ | $508$ | $1032$ |
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 $\F_{2^{2}}$.
Endomorphism algebra over $\F_{2}$The isogeny class factors as 1.2.a $\times$ 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 $\F_{2^{2}}$ is 1.4.d $\times$ 1.4.e. The endomorphism algebra for each factor is:
|
Base change
This is a primitive isogeny class.