Invariants
Base field: | $\F_{3}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 3 x + 3 x^{2} )( 1 - x + 2 x^{2} - 3 x^{3} + 9 x^{4} )$ |
$1 - 4 x + 8 x^{2} - 12 x^{3} + 24 x^{4} - 36 x^{5} + 27 x^{6}$ | |
Frobenius angles: | $\pm0.166666666667$, $\pm0.235082516458$, $\pm0.648854628963$ |
Angle rank: | $2$ (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: | $2$ |
Slopes: | $[0, 0, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $8$ | $896$ | $17024$ | $792064$ | $19535848$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $0$ | $10$ | $24$ | $114$ | $320$ | $748$ | $2240$ | $6626$ | $19176$ | $58570$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which 1 is hyperelliptic), and hence is principally polarizable:
- $y^2=2x^8+2x^7+2x^6+x^5+x^3+2x^2+2x+2$
- $x^4+2x^3z+x^2z^2+xy^2z+y^4+z^4=0$
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.ad $\times$ 2.3.ab_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 $\F_{3^{6}}$ is 1.729.cc $\times$ 2.729.abk_bas. 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 $\times$ 2.9.d_q. 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 $\times$ 2.27.ae_ak. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.