Invariants
Base field: | $\F_{7}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 + x + 7 x^{2} )( 1 - 4 x + 7 x^{2} )^{2}$ |
$1 - 7 x + 29 x^{2} - 82 x^{3} + 203 x^{4} - 343 x^{5} + 343 x^{6}$ | |
Frobenius angles: | $\pm0.227185525829$, $\pm0.227185525829$, $\pm0.560518859162$ |
Angle rank: | $1$ (numerical) |
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: | $3$ |
Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $144$ | $145152$ | $42928704$ | $14522167296$ | $4943984372784$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $1$ | $59$ | $364$ | $2519$ | $17491$ | $118508$ | $820765$ | $5756495$ | $40341028$ | $282420179$ |
Jacobians and polarizations
This isogeny class contains a Jacobian, and hence is principally polarizable.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7^{6}}$.
Endomorphism algebra over $\F_{7}$The isogeny class factors as 1.7.ae 2 $\times$ 1.7.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 $\F_{7^{6}}$ is 1.117649.la 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$ |
- Endomorphism algebra over $\F_{7^{2}}$
The base change of $A$ to $\F_{7^{2}}$ is 1.49.ac 2 $\times$ 1.49.n. The endomorphism algebra for each factor is: - 1.49.ac 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
- 1.49.n : \(\Q(\sqrt{-3}) \).
- Endomorphism algebra over $\F_{7^{3}}$
The base change of $A$ to $\F_{7^{3}}$ is 1.343.au $\times$ 1.343.u 2 . The endomorphism algebra for each factor is: - 1.343.au : \(\Q(\sqrt{-3}) \).
- 1.343.u 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
Base change
This is a primitive isogeny class.