Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 3 x^{2}$ |
| Frobenius angles: | $\pm0.5$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $2$ |
| Isomorphism classes: | 2 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4$ | $16$ | $28$ | $64$ | $244$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $4$ | $16$ | $28$ | $64$ | $244$ | $784$ | $2188$ | $6400$ | $19684$ | $59536$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which 0 are hyperelliptic):
- $y^2=x^3+x$
- $y^2=x^3+2 x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{2}}$.
Endomorphism algebra over $\F_{3}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \). |
| The base change of $A$ to $\F_{3^{2}}$ is the simple isogeny class 1.9.g and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$. |
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 |
|---|---|---|
| 1.3.ad | $3$ | 1.27.a |
| 1.3.d | $3$ | 1.27.a |
| 1.3.d | $6$ | 1.729.cc |