Invariants
Base field: | $\F_{3^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 10 x + 27 x^{2} )( 1 - 9 x + 27 x^{2} )$ |
$1 - 19 x + 144 x^{2} - 513 x^{3} + 729 x^{4}$ | |
Frobenius angles: | $\pm0.0877398280459$, $\pm0.166666666667$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $342$ | $480852$ | $383719896$ | $282467852064$ | $205964488547802$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $9$ | $657$ | $19494$ | $531513$ | $14354019$ | $387463122$ | $10460602161$ | $282430697841$ | $7625601845298$ | $205891144341057$ |
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 $\F_{3^{18}}$.
Endomorphism algebra over $\F_{3^{3}}$The isogeny class factors as 1.27.ak $\times$ 1.27.aj 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 $\F_{3^{18}}$ is 1.387420489.evq $\times$ 1.387420489.cggc. The endomorphism algebra for each factor is:
|
- Endomorphism algebra over $\F_{3^{6}}$
The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu $\times$ 1.729.abb. The endomorphism algebra for each factor is: - Endomorphism algebra over $\F_{3^{9}}$
The base change of $A$ to $\F_{3^{9}}$ is 1.19683.ahi $\times$ 1.19683.a. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.