Invariants
Base field: | $\F_{7}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 4 x + 7 x^{2} )( 1 - 5 x + 7 x^{2} )^{2}$ |
$1 - 14 x + 86 x^{2} - 296 x^{3} + 602 x^{4} - 686 x^{5} + 343 x^{6}$ | |
Frobenius angles: | $\pm0.106147807505$, $\pm0.106147807505$, $\pm0.227185525829$ |
Angle rank: | $1$ (numerical) |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $36$ | $73008$ | $38211264$ | $14126463936$ | $4829415508116$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-6$ | $26$ | $324$ | $2450$ | $17094$ | $118508$ | $825546$ | $5769314$ | $40366188$ | $282519146$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
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.af 2 $\times$ 1.7.ae 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.al 2 $\times$ 1.49.ac. The endomorphism algebra for each factor is: - 1.49.al 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
- 1.49.ac : \(\Q(\sqrt{-3}) \).
- Endomorphism algebra over $\F_{7^{3}}$
The base change of $A$ to $\F_{7^{3}}$ is 1.343.au 2 $\times$ 1.343.u. The endomorphism algebra for each factor is: - 1.343.au 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
- 1.343.u : \(\Q(\sqrt{-3}) \).
Base change
This is a primitive isogeny class.