Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + 6 x^{2} - x^{3} + 18 x^{4} + 27 x^{6}$ |
| Frobenius angles: | $\pm0.317465802914$, $\pm0.531966122430$, $\pm0.645828817189$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.6370731.1 |
| Galois group: | $A_4\times C_2$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $51$ | $2703$ | $16983$ | $532491$ | $16316481$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $4$ | $22$ | $25$ | $82$ | $274$ | $673$ | $2062$ | $6610$ | $19924$ | $59662$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which 0 are hyperelliptic):
- $2 x^4+x^3 y+x^3 z+2 x y^2 z+x z^3+y^3 z=0$
- $2 x^4+2 x^2 y^2+x^2 z^2+2 x y^2 z+x z^3+y^3 z=0$
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 6.0.6370731.1. |
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.a_g_b | $2$ | 3.9.m_cu_kj |