Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $4$ |
| L-polynomial: | $1 - 5 x + 12 x^{2} - 25 x^{3} + 49 x^{4} - 75 x^{5} + 108 x^{6} - 135 x^{7} + 81 x^{8}$ |
| Frobenius angles: | $\pm0.0766501895191$, $\pm0.123349810481$, $\pm0.476650189519$, $\pm0.676650189519$ |
| Angle rank: | $1$ (numerical) |
| Number field: | 8.0.37515625.1 |
| Galois group: | $C_4\times C_2$ |
| Jacobians: | $0$ |
| Isomorphism classes: | 2 |
This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular.
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $11$ | $5401$ | $257741$ | $35219921$ | $3529666921$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-1$ | $9$ | $8$ | $65$ | $244$ | $756$ | $2393$ | $6689$ | $19664$ | $60494$ |
Jacobians and polarizations
This isogeny class is not principally polarizable, and therefore does not contain a Jacobian.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{10}}$.
Endomorphism algebra over $\F_{3}$| The endomorphism algebra of this simple isogeny class is 8.0.37515625.1. |
| The base change of $A$ to $\F_{3^{10}}$ is 1.59049.nx 4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-35}) \)$)$ |
- Endomorphism algebra over $\F_{3^{2}}$
The base change of $A$ to $\F_{3^{2}}$ is the simple isogeny class 4.9.ab_ai_r_cd and its endomorphism algebra is 8.0.37515625.1. - Endomorphism algebra over $\F_{3^{5}}$
The base change of $A$ to $\F_{3^{5}}$ is 2.243.a_nx 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{5}, \sqrt{-7})\)$)$
Base change
This is a primitive isogeny class.