Invariants
| Base field: | $\F_{3^{3}}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 - 9 x + 27 x^{2}$ |
| Frobenius angles: | $\pm0.166666666667$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $1$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $19$ | $703$ | $19684$ | $532171$ | $14355469$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $19$ | $703$ | $19684$ | $532171$ | $14355469$ | $387459856$ | $10460530351$ | $282430067923$ | $7625597484988$ | $205891117745743$ |
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 $\F_{3^{18}}$.
Endomorphism algebra over $\F_{3^{3}}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \). |
| The base change of $A$ to $\F_{3^{18}}$ is the simple isogeny class 1.387420489.cggc and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$. |
- Endomorphism algebra over $\F_{3^{6}}$
The base change of $A$ to $\F_{3^{6}}$ is the simple isogeny class 1.729.abb and its endomorphism algebra is \(\Q(\sqrt{-3}) \). - Endomorphism algebra over $\F_{3^{9}}$
The base change of $A$ to $\F_{3^{9}}$ is the simple isogeny class 1.19683.a and its endomorphism algebra is \(\Q(\sqrt{-3}) \).
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 |
|---|---|---|
| 1.27.j | $2$ | 1.729.abb |
| 1.27.a | $3$ | (not in LMFDB) |
| 1.27.j | $3$ | (not in LMFDB) |