Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 3 x + 9 x^{2} - 15 x^{3} + 27 x^{4} - 27 x^{5} + 27 x^{6}$ |
| Frobenius angles: | $\pm0.239076736162$, $\pm0.372844462505$, $\pm0.581699150202$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.465831.1 |
| Galois group: | $A_4\times C_2$ |
| Jacobians: | $1$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $19$ | $2071$ | $27037$ | $598519$ | $15914989$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $1$ | $19$ | $37$ | $91$ | $271$ | $703$ | $2080$ | $6643$ | $19765$ | $58159$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is hyperelliptic):
- $y^2=2 x^8+2 x^7+x^6+2 x^5+x^4+x^2+x+2$
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 6.0.465831.1. |
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.d_j_p | $2$ | 3.9.j_bt_fx |