Invariants
Base field: | $\F_{2}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 2 x + 2 x^{2} )( 1 - 2 x^{2} + 4 x^{4} )$ |
$1 - 2 x + 4 x^{3} - 8 x^{5} + 8 x^{6}$ | |
Frobenius angles: | $\pm0.166666666667$, $\pm0.250000000000$, $\pm0.833333333333$ |
Angle rank: | $0$ (numerical) |
Jacobians: | $1$ |
Isomorphism classes: | 5 |
This isogeny class is not simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is supersingular.
$p$-rank: | $0$ |
Slopes: | $[1/2, 1/2, 1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $3$ | $45$ | $1053$ | $11025$ | $40713$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $1$ | $1$ | $13$ | $33$ | $41$ | $97$ | $113$ | $257$ | $481$ | $961$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is not hyperelliptic), and hence is principally polarizable:
- $x^4+x^3y+x^3z+xz^3+y^4=0$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{24}}$.
Endomorphism algebra over $\F_{2}$The isogeny class factors as 1.2.ac $\times$ 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 $\F_{2^{24}}$ is 1.16777216.amdc 3 and its endomorphism algebra is $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$. |
- Endomorphism algebra over $\F_{2^{2}}$
The base change of $A$ to $\F_{2^{2}}$ is 1.4.ac 2 $\times$ 1.4.a. The endomorphism algebra for each factor is: - 1.4.ac 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
- 1.4.a : \(\Q(\sqrt{-1}) \).
- Endomorphism algebra over $\F_{2^{3}}$
The base change of $A$ to $\F_{2^{3}}$ is 1.8.a 2 $\times$ 1.8.e. The endomorphism algebra for each factor is: - 1.8.a 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
- 1.8.e : \(\Q(\sqrt{-1}) \).
- Endomorphism algebra over $\F_{2^{4}}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.e 2 $\times$ 1.16.i. The endomorphism algebra for each factor is: - 1.16.e 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
- 1.16.i : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- Endomorphism algebra over $\F_{2^{6}}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.a $\times$ 1.64.q 2 . The endomorphism algebra for each factor is: - 1.64.a : \(\Q(\sqrt{-1}) \).
- 1.64.q 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- Endomorphism algebra over $\F_{2^{8}}$
The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg $\times$ 1.256.q 2 . The endomorphism algebra for each factor is: - 1.256.abg : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 1.256.q 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
- Endomorphism algebra over $\F_{2^{12}}$
The base change of $A$ to $\F_{2^{12}}$ is 1.4096.aey 2 $\times$ 1.4096.ey. The endomorphism algebra for each factor is: - 1.4096.aey 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 1.4096.ey : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
Base change
This is a primitive isogeny class.