Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + x + 3 x^{2} + 8 x^{3} + 9 x^{4} + 9 x^{5} + 27 x^{6}$ |
| Frobenius angles: | $\pm0.330193618908$, $\pm0.466552925350$, $\pm0.857957776457$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.76139264.1 |
| Galois group: | $S_4\times C_2$ |
| Jacobians: | $2$ |
| Cyclic group of points: | yes |
This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $58$ | $1276$ | $36424$ | $515504$ | $11798998$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $5$ | $15$ | $44$ | $79$ | $195$ | $768$ | $2105$ | $6671$ | $19556$ | $59695$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which 1 is hyperelliptic):
- $y^2=2 x^7+x^6+2 x^5+x^2+2 x+2$
- $2 x^3 z+2 x^2 y^2+2 x^2 y z+x^2 z^2+x y^3+x z^3+y^2 z^2=0$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3}$.
Endomorphism algebra over $\F_{3}$| The endomorphism algebra of this simple isogeny class is 6.0.76139264.1. |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 3.3.ab_d_ai | $2$ | 3.9.f_l_ba |