Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $4$ |
| L-polynomial: | $1 - 9 x^{4} + 81 x^{8}$ |
| Frobenius angles: | $\pm0.0833333333333$, $\pm0.416666666667$, $\pm0.583333333333$, $\pm0.916666666667$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\zeta_{24})\) |
| Galois group: | $C_2^3$ |
| Isomorphism classes: | 41 |
This isogeny class is simple but not geometrically simple, primitive, not ordinary, and supersingular.
Newton polygon
This isogeny class is supersingular.
| $p$-rank: | $0$ |
| Slopes: | $[1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $73$ | $5329$ | $532900$ | $28398241$ | $3486725353$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $4$ | $10$ | $28$ | $46$ | $244$ | $730$ | $2188$ | $6886$ | $19684$ | $59050$ |
Jacobians and polarizations
It is unknown whether this isogeny class contains a Jacobian or whether it is principally polarizable.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{12}}$.
Endomorphism algebra over $\F_{3}$| The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{24})\). |
| The base change of $A$ to $\F_{3^{12}}$ is 1.531441.cec 4 and its endomorphism algebra is $\mathrm{M}_{4}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$. |
- Endomorphism algebra over $\F_{3^{2}}$
The base change of $A$ to $\F_{3^{2}}$ is 2.9.a_aj 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\zeta_{12})\)$)$ - Endomorphism algebra over $\F_{3^{3}}$
The base change of $A$ to $\F_{3^{3}}$ is 2.27.a_a 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(i, \sqrt{6})\)$)$ - Endomorphism algebra over $\F_{3^{4}}$
The base change of $A$ to $\F_{3^{4}}$ is 1.81.aj 4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-3}) \)$)$ - Endomorphism algebra over $\F_{3^{6}}$
The base change of $A$ to $\F_{3^{6}}$ is 1.729.a 4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-1}) \)$)$
Base change
This is a primitive isogeny class.