Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 3 x )^{2}( 1 - 3 x + 9 x^{2} )$ |
$1 - 9 x + 36 x^{2} - 81 x^{3} + 81 x^{4}$ | |
Frobenius angles: | $0$, $0$, $\pm0.333333333333$ |
Angle rank: | $0$ (numerical) |
Jacobians: | $0$ |
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: | $[1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $28$ | $5824$ | $529984$ | $42515200$ | $3443973148$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $1$ | $73$ | $730$ | $6481$ | $58321$ | $528526$ | $4776409$ | $43040161$ | $387420490$ | $3486725353$ |
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 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 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, 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. The endomorphism algebra for each factor is: - 1.81.as : 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 $\times$ 1.729.cc. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.