Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 3 x )^{2}( 1 - 3 x + 9 x^{2} )^{2}$ |
$1 - 12 x + 72 x^{2} - 270 x^{3} + 648 x^{4} - 972 x^{5} + 729 x^{6}$ | |
Frobenius angles: | $0$, $0$, $\pm0.333333333333$, $\pm0.333333333333$ |
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: | $[1/2, 1/2, 1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $196$ | $529984$ | $415507456$ | $282428473600$ | $202529728914436$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-2$ | $82$ | $784$ | $6562$ | $58078$ | $527068$ | $4774222$ | $43046722$ | $387459856$ | $3486784402$ |
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 $\F_{3^{12}}$.
Endomorphism algebra over $\F_{3^{2}}$The isogeny class factors as 1.9.ag $\times$ 1.9.ad 2 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 $\F_{3^{12}}$ is 1.531441.acec 3 and its endomorphism algebra is $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$. |
- Endomorphism algebra over $\F_{3^{4}}$
The base change of $A$ to $\F_{3^{4}}$ is 1.81.as $\times$ 1.81.j 2 . The endomorphism algebra for each factor is: - 1.81.as : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
- 1.81.j 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
- Endomorphism algebra over $\F_{3^{6}}$
The base change of $A$ to $\F_{3^{6}}$ is 1.729.acc $\times$ 1.729.cc 2 . The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.