Invariants
Base field: | $\F_{3}$ |
Dimension: | $4$ |
L-polynomial: | $( 1 - x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{3}$ |
$1 - 10 x + 48 x^{2} - 144 x^{3} + 297 x^{4} - 432 x^{5} + 432 x^{6} - 270 x^{7} + 81 x^{8}$ | |
Frobenius angles: | $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.406785250661$ |
Angle rank: | $1$ (numerical) |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $3$ | $5145$ | $790272$ | $56517825$ | $4239234843$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-6$ | $6$ | $36$ | $102$ | $294$ | $882$ | $2514$ | $6918$ | $19548$ | $57846$ |
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 $\F_{3^{6}}$.
Endomorphism algebra over $\F_{3}$The isogeny class factors as 1.3.ad 3 $\times$ 1.3.ab 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.ak $\times$ 1.729.cc 3 . 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 3 $\times$ 1.9.f. The endomorphism algebra for each factor is: - 1.9.ad 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
- 1.9.f : \(\Q(\sqrt{-11}) \).
- Endomorphism algebra over $\F_{3^{3}}$
The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 3 $\times$ 1.27.i. The endomorphism algebra for each factor is: - 1.27.a 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
- 1.27.i : \(\Q(\sqrt{-11}) \).
Base change
This is a primitive isogeny class.