Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $4$ |
| L-polynomial: | $1 - x + 6 x^{2} - 7 x^{3} + 20 x^{4} - 21 x^{5} + 54 x^{6} - 27 x^{7} + 81 x^{8}$ |
| Frobenius angles: | $\pm0.200745078075$, $\pm0.430271667139$, $\pm0.541713501460$, $\pm0.707067623389$ |
| Angle rank: | $4$ (numerical) |
| Number field: | 8.0.4897800252736.1 |
| Galois group: | $C_2 \wr S_4$ |
| Jacobians: | $3$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $106$ | $23108$ | $473608$ | $44090064$ | $3765114806$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $3$ | $21$ | $24$ | $85$ | $263$ | $786$ | $2327$ | $6381$ | $19338$ | $58881$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which 0 are hyperelliptic):
- $x y+t^2=x y^2+y^3+x^2 z-y^2 z-x z^2-z^3+y^2 t-y z t-z^2 t=0$
- $x y+t^2=x y^2+y^3+x^2 z+x y z+x z^2-y z^2-z^3+x y t+y^2 t+y z t+z^2 t=0$
- $x t-y z=x^2 y+y^3+x^2 z+z^3+x y t+x z t+z^2 t+x t^2-y t^2+t^3=0$
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 8.0.4897800252736.1. |
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.b_g_h_u | $2$ | (not in LMFDB) |