Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 3 x + 9 x^{2} )( 1 - 2 x + 9 x^{2} )$ |
$1 - 5 x + 24 x^{2} - 45 x^{3} + 81 x^{4}$ | |
Frobenius angles: | $\pm0.333333333333$, $\pm0.391826552031$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $56$ | $8736$ | $608384$ | $43365504$ | $3444208376$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $105$ | $830$ | $6609$ | $58325$ | $529326$ | $4783805$ | $43065249$ | $387456590$ | $3486729225$ |
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.ad $\times$ 1.9.ac 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.bu $\times$ 1.729.cc. The endomorphism algebra for each factor is:
|
Base change
This is a primitive isogeny class.