Invariants
Base field: | $\F_{3}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - x + 3 x^{2} )( 1 + x + 3 x^{2} )$ |
$1 + 5 x^{2} + 9 x^{4}$ | |
Frobenius angles: | $\pm0.406785250661$, $\pm0.593214749339$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $2$ |
Isomorphism classes: | 4 |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $15$ | $225$ | $720$ | $5625$ | $58575$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $20$ | $28$ | $68$ | $244$ | $710$ | $2188$ | $6788$ | $19684$ | $58100$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+x^5+x^3+x+2$
- $y^2=x^6+2x^5+2x^3+2x+1$
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.b 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 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$ |
Base change
This is a primitive isogeny class.