Invariants
Base field: | $\F_{3}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - x + 3 x^{2} )( 1 + 3 x + 3 x^{2} )$ |
$1 + 2 x + 3 x^{2} + 6 x^{3} + 9 x^{4}$ | |
Frobenius angles: | $\pm0.406785250661$, $\pm0.833333333333$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $2$ |
Isomorphism classes: | 4 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $21$ | $105$ | $1008$ | $6825$ | $46221$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $12$ | $36$ | $84$ | $186$ | $774$ | $2190$ | $6756$ | $19548$ | $58332$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=x^6+x^4+2x^3+2x+1$
- $y^2=x^6+x^5+2x^4+2x^2+x+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{6}}$.
Endomorphism algebra over $\F_{3}$The isogeny class factors as 1.3.ab $\times$ 1.3.d 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 $\F_{3^{6}}$ is 1.729.ak $\times$ 1.729.cc. The endomorphism algebra for each factor is:
|
- Endomorphism algebra over $\F_{3^{2}}$
The base change of $A$ to $\F_{3^{2}}$ is 1.9.ad $\times$ 1.9.f. The endomorphism algebra for each factor is: - Endomorphism algebra over $\F_{3^{3}}$
The base change of $A$ to $\F_{3^{3}}$ is 1.27.a $\times$ 1.27.i. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.