Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - x + 5 x^{2} - 4 x^{3} + 15 x^{4} - 9 x^{5} + 27 x^{6}$ |
| Frobenius angles: | $\pm0.263563453386$, $\pm0.456615573474$, $\pm0.675390095728$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.66703808.1 |
| Galois group: | $S_4\times C_2$ |
| Jacobians: | $1$ |
| Isomorphism classes: | 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $34$ | $2108$ | $20536$ | $615536$ | $14969894$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $3$ | $19$ | $30$ | $95$ | $253$ | $700$ | $2243$ | $6383$ | $19038$ | $59699$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is hyperelliptic):
- $y^2=x^8+x^6+x^5+x^4+2 x^3+2 x^2+x+2$
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 6.0.66703808.1. |
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.b_f_e | $2$ | 3.9.j_bv_go |