Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + 3 x + 6 x^{2} + 12 x^{3} + 18 x^{4} + 27 x^{5} + 27 x^{6}$ |
| Frobenius angles: | $\pm0.354495973058$, $\pm0.609416820172$, $\pm0.922323234158$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.5604552.1 |
| Galois group: | $S_4\times C_2$ |
| Jacobians: | $2$ |
| Cyclic group of points: | yes |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
| $p$-rank: | $0$ |
| Slopes: | $[1/3, 1/3, 1/3, 2/3, 2/3, 2/3]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $94$ | $940$ | $26038$ | $488800$ | $15432544$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $7$ | $13$ | $37$ | $73$ | $262$ | $649$ | $2107$ | $6913$ | $19765$ | $58888$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which 1 is hyperelliptic):
- $y^2=x^8+2 x^7+x$
- $x^4+2 x^3 y+x^2 y^2+x z^3+y^3 z=0$
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 6.0.5604552.1. |
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.ad_g_am | $2$ | 3.9.d_a_abk |