Invariants
Base field: | $\F_{2}$ |
Dimension: | $4$ |
L-polynomial: | $( 1 - x + 2 x^{2} )( 1 - 3 x + 2 x^{2} + x^{3} + 4 x^{4} - 12 x^{5} + 8 x^{6} )$ |
$1 - 4 x + 7 x^{2} - 7 x^{3} + 7 x^{4} - 14 x^{5} + 28 x^{6} - 32 x^{7} + 16 x^{8}$ | |
Frobenius angles: | $\pm0.0992589862044$, $\pm0.186455299510$, $\pm0.384973271919$, $\pm0.757883870938$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $0$ |
Isomorphism classes: | 3 |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $4$ |
Slopes: | $[0, 0, 0, 0, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2$ | $232$ | $4214$ | $130384$ | $766942$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-1$ | $3$ | $8$ | $27$ | $24$ | $78$ | $181$ | $283$ | $575$ | $958$ |
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 $\F_{2^{7}}$.
Endomorphism algebra over $\F_{2}$The isogeny class factors as 1.2.ab $\times$ 3.2.ad_c_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 $\F_{2^{7}}$ is 1.128.n 4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-7}) \)$)$ |
Base change
This is a primitive isogeny class.