Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 4 x + 9 x^{2} )( 1 - 8 x + 32 x^{2} - 72 x^{3} + 81 x^{4} )$ |
$1 - 12 x + 73 x^{2} - 272 x^{3} + 657 x^{4} - 972 x^{5} + 729 x^{6}$ | |
Frobenius angles: | $\pm0.141826552031$, $\pm0.267720472801$, $\pm0.358173447969$ |
Angle rank: | $2$ (numerical) |
Isomorphism classes: | 22 |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $3$ |
Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $204$ | $554064$ | $434556108$ | $292368491520$ | $206409655841964$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-2$ | $84$ | $814$ | $6788$ | $59198$ | $530964$ | $4783406$ | $43058812$ | $387464926$ | $3486846804$ |
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 $\F_{3^{8}}$.
Endomorphism algebra over $\F_{3^{2}}$The isogeny class factors as 1.9.ae $\times$ 2.9.ai_bg 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 $\F_{3^{8}}$ is 1.6561.bi 2 $\times$ 1.6561.gc. The endomorphism algebra for each factor is:
|
- Endomorphism algebra over $\F_{3^{4}}$
The base change of $A$ to $\F_{3^{4}}$ is 1.81.c $\times$ 2.81.a_bi. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.