Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 3 x )^{2}( 1 + 9 x^{2} )( 1 + 2 x + 9 x^{2} )$ |
$1 - 4 x + 15 x^{2} - 72 x^{3} + 135 x^{4} - 324 x^{5} + 729 x^{6}$ | |
Frobenius angles: | $0$, $0$, $\pm0.5$, $\pm0.608173447969$ |
Angle rank: | $1$ (numerical) |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $480$ | $614400$ | $337540320$ | $267386880000$ | $205873812434400$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $96$ | $630$ | $6204$ | $59046$ | $530784$ | $4775574$ | $43032444$ | $387384390$ | $3486670176$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but it is unknown whether it contains 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.ag $\times$ 1.9.a $\times$ 1.9.c 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.agg 2 $\times$ 1.6561.abi. The endomorphism algebra for each factor is:
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- Endomorphism algebra over $\F_{3^{4}}$
The base change of $A$ to $\F_{3^{4}}$ is 1.81.as $\times$ 1.81.o $\times$ 1.81.s. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.