Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 4 x + 9 x^{2} )( 1 - 3 x + 9 x^{2} )$ |
$1 - 7 x + 30 x^{2} - 63 x^{3} + 81 x^{4}$ | |
Frobenius angles: | $\pm0.267720472801$, $\pm0.333333333333$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $0$ |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $42$ | $7644$ | $606816$ | $44640960$ | $3486431802$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $3$ | $93$ | $828$ | $6801$ | $59043$ | $529506$ | $4776747$ | $43041441$ | $387448812$ | $3486905853$ |
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 $\F_{3^{6}}$.
Endomorphism algebra over $\F_{3^{2}}$The isogeny class factors as 1.9.ae $\times$ 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 $\F_{3^{6}}$ is 1.729.bs $\times$ 1.729.cc. The endomorphism algebra for each factor is:
|
Base change
This is a primitive isogeny class.