Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + 2 x + 5 x^{2} + 5 x^{3} + 15 x^{4} + 18 x^{5} + 27 x^{6}$ |
| Frobenius angles: | $\pm0.311296029718$, $\pm0.638747551283$, $\pm0.753048203555$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.25119239.1 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $73$ | $1679$ | $14308$ | $807599$ | $14204048$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $16$ | $21$ | $116$ | $241$ | $637$ | $2162$ | $6548$ | $19929$ | $59291$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which 1 is hyperelliptic):
- $y^2=x^8+2 x^7+2 x^6+2 x^5+x^4+x^2+x+1$
- $2 x^3 y+2 x^2 y^2+2 x^2 y z+x y^2 z+x y z^2+x z^3+y^3 z=0$
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 6.0.25119239.1. |
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.ac_f_af | $2$ | 3.9.g_bj_ed |