Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 3 x + 3 x^{2}$ |
| Frobenius angles: | $\pm0.833333333333$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $1$ |
| Isomorphism classes: | 1 |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is supersingular.
| $p$-rank: | $0$ |
| Slopes: | $[1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $7$ | $7$ | $28$ | $91$ | $217$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $7$ | $7$ | $28$ | $91$ | $217$ | $784$ | $2107$ | $6643$ | $19684$ | $58807$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{6}}$.
Endomorphism algebra over $\F_{3}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \). |
| The base change of $A$ to $\F_{3^{6}}$ is the simple isogeny class 1.729.cc and its endomorphism algebra 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 the simple isogeny class 1.9.ad and its endomorphism algebra is \(\Q(\sqrt{-3}) \). - Endomorphism algebra over $\F_{3^{3}}$
The base change of $A$ to $\F_{3^{3}}$ is the simple isogeny class 1.27.a and its endomorphism algebra is \(\Q(\sqrt{-3}) \).
Base change
This is a primitive isogeny class.