Invariants
| Base field: | $\F_{7^{3}}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 343 x^{2}$ |
| Frobenius angles: | $\pm0.5$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{-7}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $2$ |
This isogeny class is simple and geometrically simple, not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $344$ | $118336$ | $40353608$ | $13841051904$ | $4747561509944$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $344$ | $118336$ | $40353608$ | $13841051904$ | $4747561509944$ | $1628413678617664$ | $558545864083284008$ | $191581231352883840000$ | $65712362363534280139544$ | $22539340290701753210883136$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7^{6}}$.
Endomorphism algebra over $\F_{7^{3}}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-7}) \). |
| The base change of $A$ to $\F_{7^{6}}$ is the simple isogeny class 1.117649.bak and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $7$ and $\infty$. |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{7^{3}}$.
| Subfield | Primitive Model |
| $\F_{7}$ | 1.7.a |
Twists
This isogeny class has no twists.