Invariants
Base field: | $\F_{2}$ |
Dimension: | $4$ |
L-polynomial: | $( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )( 1 - x - x^{2} - 2 x^{3} + 4 x^{4} )$ |
$1 - 4 x + 7 x^{2} - 10 x^{3} + 15 x^{4} - 20 x^{5} + 28 x^{6} - 32 x^{7} + 16 x^{8}$ | |
Frobenius angles: | $\pm0.0516399385854$, $\pm0.123548644961$, $\pm0.456881978294$, $\pm0.718306605252$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $0$ |
Isomorphism classes: | 4 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1$ | $133$ | $1216$ | $44289$ | $721711$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-1$ | $3$ | $-1$ | $11$ | $19$ | $69$ | $181$ | $243$ | $503$ | $1143$ |
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 $\F_{2^{6}}$.
Endomorphism algebra over $\F_{2}$The isogeny class factors as 2.2.ad_f $\times$ 2.2.ab_ab 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 $\F_{2^{6}}$ is 1.64.aj 2 $\times$ 1.64.l 2 . The endomorphism algebra for each factor is:
|
- Endomorphism algebra over $\F_{2^{2}}$
The base change of $A$ to $\F_{2^{2}}$ is 2.4.ad_f $\times$ 2.4.b_ad. The endomorphism algebra for each factor is: - Endomorphism algebra over $\F_{2^{3}}$
The base change of $A$ to $\F_{2^{3}}$ is 1.8.af 2 $\times$ 2.8.a_l. The endomorphism algebra for each factor is: - 1.8.af 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
- 2.8.a_l : \(\Q(\sqrt{-3}, \sqrt{5})\).
Base change
This is a primitive isogeny class.