Invariants
Base field: | $\F_{2}$ |
Dimension: | $4$ |
L-polynomial: | $( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )( 1 - 2 x^{2} + 4 x^{4} )$ |
$1 - 3 x + 3 x^{2} - 2 x^{4} + 12 x^{6} - 24 x^{7} + 16 x^{8}$ | |
Frobenius angles: | $\pm0.123548644961$, $\pm0.166666666667$, $\pm0.456881978294$, $\pm0.833333333333$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $1$ |
Isomorphism classes: | 4 |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $3$ | $171$ | $6156$ | $75411$ | $954273$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $0$ | $2$ | $9$ | $18$ | $30$ | $119$ | $168$ | $306$ | $513$ | $1022$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is not hyperelliptic), and hence is principally polarizable:
- $x^2+xy+y^2+zt=y^3+xyz+z^3+x^2t+t^3=0$
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.a_ac 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.l 2 $\times$ 1.64.q 2 . 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.ac 2 $\times$ 2.4.b_ad. The endomorphism algebra for each factor is: - 1.4.ac 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
- 2.4.b_ad : \(\Q(\sqrt{-3}, \sqrt{5})\).
- Endomorphism algebra over $\F_{2^{3}}$
The base change of $A$ to $\F_{2^{3}}$ is 1.8.a 2 $\times$ 2.8.a_l. The endomorphism algebra for each factor is: - 1.8.a 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
- 2.8.a_l : \(\Q(\sqrt{-3}, \sqrt{5})\).
Base change
This is a primitive isogeny class.