Invariants
Base field: | $\F_{2^{2}}$ |
Dimension: | $4$ |
L-polynomial: | $( 1 - 2 x )^{4}( 1 - 2 x + 4 x^{2} - 8 x^{3} + 16 x^{4} )$ |
$1 - 10 x + 44 x^{2} - 120 x^{3} + 256 x^{4} - 480 x^{5} + 704 x^{6} - 640 x^{7} + 256 x^{8}$ | |
Frobenius angles: | $0$, $0$, $0$, $0$, $\pm0.200000000000$, $\pm0.600000000000$ |
Angle rank: | $0$ (numerical) |
This isogeny class is not simple, primitive, not ordinary, and supersingular. It is principally polarizable.
Newton polygon
This isogeny class is supersingular.
$p$-rank: | $0$ |
Slopes: | $[1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $11$ | $27621$ | $8742041$ | $3538940625$ | $1095222947841$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-5$ | $5$ | $25$ | $209$ | $1025$ | $3905$ | $15745$ | $64769$ | $259585$ | $1040385$ |
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 $\F_{2^{20}}$.
Endomorphism algebra over $\F_{2^{2}}$The isogeny class factors as 1.4.ae 2 $\times$ 2.4.ac_e 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 $\F_{2^{20}}$ is 1.1048576.adau 4 and its endomorphism algebra is $\mathrm{M}_{4}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$. |
- Endomorphism algebra over $\F_{2^{4}}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.ai 2 $\times$ 2.16.e_q. The endomorphism algebra for each factor is: - 1.16.ai 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 2.16.e_q : \(\Q(\zeta_{5})\).
- Endomorphism algebra over $\F_{2^{10}}$
The base change of $A$ to $\F_{2^{10}}$ is 1.1024.acm 2 $\times$ 1.1024.cm 2 . The endomorphism algebra for each factor is: - 1.1024.acm 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 1.1024.cm 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
Base change
This is a primitive isogeny class.