Invariants
Base field: | $\F_{3}$ |
Dimension: | $4$ |
L-polynomial: | $( 1 - 2 x + 3 x^{2} )^{2}( 1 - 4 x + 8 x^{2} - 12 x^{3} + 9 x^{4} )$ |
$1 - 8 x + 34 x^{2} - 96 x^{3} + 194 x^{4} - 288 x^{5} + 306 x^{6} - 216 x^{7} + 81 x^{8}$ | |
Frobenius angles: | $\pm0.0540867239847$, $\pm0.304086723985$, $\pm0.304086723985$, $\pm0.445913276015$ |
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: | $4$ |
Slopes: | $[0, 0, 0, 0, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $8$ | $9792$ | $903944$ | $42614784$ | $2886151048$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-4$ | $14$ | $44$ | $82$ | $196$ | $638$ | $2068$ | $6426$ | $19772$ | $60014$ |
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 $\F_{3^{8}}$.
Endomorphism algebra over $\F_{3}$The isogeny class factors as 1.3.ac 2 $\times$ 2.3.ae_i 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 $\F_{3^{8}}$ is 1.6561.abi 4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-2}) \)$)$ |
- Endomorphism algebra over $\F_{3^{2}}$
The base change of $A$ to $\F_{3^{2}}$ is 1.9.c 2 $\times$ 2.9.a_ao. The endomorphism algebra for each factor is: - 1.9.c 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
- 2.9.a_ao : \(\Q(\zeta_{8})\).
- Endomorphism algebra over $\F_{3^{4}}$
The base change of $A$ to $\F_{3^{4}}$ is 1.81.ao 2 $\times$ 1.81.o 2 . The endomorphism algebra for each factor is: - 1.81.ao 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
- 1.81.o 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
Base change
This is a primitive isogeny class.