Invariants
Base field: | $\F_{3}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 2 x + 3 x^{2} )( 1 + 3 x^{2} )$ |
$1 - 2 x + 6 x^{2} - 6 x^{3} + 9 x^{4}$ | |
Frobenius angles: | $\pm0.304086723985$, $\pm0.5$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $2$ |
Isomorphism classes: | 6 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $8$ | $192$ | $1064$ | $6144$ | $59048$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $2$ | $18$ | $38$ | $78$ | $242$ | $738$ | $2102$ | $6366$ | $19874$ | $60018$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^5+x^4+2x^3+x^2+2x$
- $y^2=2x^6+2x^5+x^4+x^3+x^2+2x+2$
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.ac $\times$ 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.c $\times$ 1.9.g. The endomorphism algebra for each factor is:
|
Base change
This is a primitive isogeny class.