Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 5 x + 9 x^{2} )( 1 - 3 x + 9 x^{2} )$ |
$1 - 8 x + 33 x^{2} - 72 x^{3} + 81 x^{4}$ | |
Frobenius angles: | $\pm0.186429498677$, $\pm0.333333333333$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $1$ |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $35$ | $6825$ | $580160$ | $44342025$ | $3500486675$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $2$ | $84$ | $794$ | $6756$ | $59282$ | $531342$ | $4783298$ | $43053636$ | $387438986$ | $3486735924$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2=2ax^6+ax^5+2ax^4+2x^3+2ax^2+ax+2a$
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.af $\times$ 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.k $\times$ 1.729.cc. The endomorphism algebra for each factor is:
|
Base change
This is a primitive isogeny class.