Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 4 x + 8 x^{2} - 13 x^{3} + 24 x^{4} - 36 x^{5} + 27 x^{6}$ |
| Frobenius angles: | $\pm0.102762435325$, $\pm0.278353759721$, $\pm0.643265352440$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.10816643.1 |
| Galois group: | $A_4\times C_2$ |
| Jacobians: | $1$ |
| Isomorphism classes: | 1 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $7$ | $791$ | $14623$ | $655739$ | $16743377$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $0$ | $10$ | $21$ | $98$ | $280$ | $673$ | $2156$ | $6722$ | $19740$ | $59810$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is not hyperelliptic), and hence is principally polarizable:
- $2x^4+x^3y+x^2yz+x^2z^2+xy^2z+y^4+z^4=0$
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 6.0.10816643.1. |
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.e_i_n | $2$ | 3.9.a_i_at |