Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 3 x )^{4}( 1 - 3 x + 9 x^{2} )$ |
$1 - 15 x + 99 x^{2} - 378 x^{3} + 891 x^{4} - 1215 x^{5} + 729 x^{6}$ | |
Frobenius angles: | $0$, $0$, $0$, $0$, $\pm0.333333333333$ |
Angle rank: | $0$ (numerical) |
This isogeny class is not simple, not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $112$ | $372736$ | $358269184$ | $272097280000$ | $201692843439472$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-5$ | $55$ | $676$ | $6319$ | $57835$ | $527068$ | $4772035$ | $43027039$ | $387381124$ | $3486607255$ |
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 2 $\times$ 1.9.ad 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 2 $\times$ 1.81.j. The endomorphism algebra for each factor is: - 1.81.as 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
- 1.81.j : \(\Q(\sqrt{-3}) \).
- Endomorphism algebra over $\F_{3^{6}}$
The base change of $A$ to $\F_{3^{6}}$ is 1.729.acc 2 $\times$ 1.729.cc. The endomorphism algebra for each factor is:
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{2}}$.
Subfield | Primitive Model |
$\F_{3}$ | 3.3.ad_ad_s |
$\F_{3}$ | 3.3.d_ad_as |