Invariants
Base field: | $\F_{3}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - x + 3 x^{2} )( 1 + 3 x^{2} )$ |
$1 - x + 6 x^{2} - 3 x^{3} + 9 x^{4}$ | |
Frobenius angles: | $\pm0.406785250661$, $\pm0.5$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $0$ |
Isomorphism classes: | 2 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $12$ | $240$ | $1008$ | $4800$ | $51972$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $3$ | $21$ | $36$ | $57$ | $213$ | $774$ | $2271$ | $6513$ | $19548$ | $59061$ |
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 $\F_{3^{2}}$.
Endomorphism algebra over $\F_{3}$The isogeny class factors as 1.3.ab $\times$ 1.3.a 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 $\F_{3^{2}}$ is 1.9.f $\times$ 1.9.g. The endomorphism algebra for each factor is:
|
Base change
This is a primitive isogeny class.