Invariants
Base field: | $\F_{3}$ |
Dimension: | $3$ |
L-polynomial: | $1 - 3 x + 8 x^{3} - 27 x^{5} + 27 x^{6}$ |
Frobenius angles: | $\pm0.0634879191729$, $\pm0.233030297526$, $\pm0.828250460517$ |
Angle rank: | $3$ (numerical) |
Number field: | 6.0.5604552.1 |
Galois group: | $S_4\times C_2$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $6$ | $300$ | $18198$ | $636000$ | $13286496$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $1$ | $1$ | $25$ | $97$ | $226$ | $781$ | $2059$ | $6481$ | $19519$ | $58696$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is not hyperelliptic), and hence is principally polarizable:
- $2x^4+x^3y+2x^3z+xyz^2+xz^3+y^4=0$
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 6.0.5604552.1. |
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.d_a_ai | $2$ | 3.9.aj_bw_agq |