Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $5$ |
| L-polynomial: | $( 1 - 3 x + 3 x^{2} )^{3}( 1 - 2 x + x^{2} - 6 x^{3} + 9 x^{4} )$ |
| $1 - 11 x + 55 x^{2} - 168 x^{3} + 369 x^{4} - 675 x^{5} + 1107 x^{6} - 1512 x^{7} + 1485 x^{8} - 891 x^{9} + 243 x^{10}$ | |
| Frobenius angles: | $\pm0.0292466093486$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.637420057318$ |
| Angle rank: | $1$ (numerical) |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $3$ | $19551$ | $7112448$ | $4166494059$ | $1160774149053$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-7$ | $-1$ | $8$ | $95$ | $323$ | $800$ | $2345$ | $6839$ | $19304$ | $57839$ |
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 $\times$ 2.3.ac_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 $\F_{3^{6}}$ is 1.729.abu 2 $\times$ 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 $\times$ 2.9.ac_af. The endomorphism algebra for each factor is: - 1.9.ad 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
- 2.9.ac_af : \(\Q(\sqrt{-2}, \sqrt{-3})\).
- Endomorphism algebra over $\F_{3^{3}}$
The base change of $A$ to $\F_{3^{3}}$ is 1.27.ak 2 $\times$ 1.27.a 3 . The endomorphism algebra for each factor is: - 1.27.ak 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
- 1.27.a 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
Base change
This is a primitive isogeny class.