Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $4$ |
| L-polynomial: | $1 + 2 x + 4 x^{2} + 8 x^{3} + 22 x^{4} + 24 x^{5} + 36 x^{6} + 54 x^{7} + 81 x^{8}$ |
| Frobenius angles: | $\pm0.280972140195$, $\pm0.400662298547$, $\pm0.735409692945$, $\pm0.820692504341$ |
| Angle rank: | $4$ (numerical) |
| Number field: | 8.0.56146817344.1 |
| Galois group: | $C_2 \wr S_4$ |
| Jacobians: | $5$ |
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: | $4$ |
| Slopes: | $[0, 0, 0, 0, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $232$ | $12992$ | $712936$ | $70052864$ | $2629335112$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $14$ | $36$ | $122$ | $176$ | $722$ | $2190$ | $6458$ | $19818$ | $58794$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which 1 is hyperelliptic):
- $y^2=2 x^{10}+x^9+x^8+2 x^7+x^6+x^3+2 x^2+2 x+1$
- $x y+t^2=y^3-y^2 z+x z^2+y z^2-z^3+x^2 t-x y t-y^2 t+x z t+y z t-z^2 t=0$
- $x y+t^2=x y^2+y^3+y^2 z+x z^2-z^3+x^2 t-y^2 t-z^2 t=0$
- $x t-y z=x y^2+x^2 z-z^3+x y t+y^2 t-z^2 t+y t^2-z t^2+t^3=0$
- $x^2+y^2+z t=y^2 z+y z^2+y^2 t+y z t+x t^2-y t^2+z t^2=0$
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 8.0.56146817344.1. |
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 |
|---|---|---|
| 4.3.ac_e_ai_w | $2$ | (not in LMFDB) |