Invariants
| Base field: | $\F_{3^{3}}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 27 x^{2}$ |
| Frobenius angles: | $\pm0.5$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}) \) |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $28$ | $784$ | $19684$ | $529984$ | $14348908$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $28$ | $784$ | $19684$ | $529984$ | $14348908$ | $387459856$ | $10460353204$ | $282428473600$ | $7625597484988$ | $205891160792464$ |
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 $\F_{3^{6}}$.
Endomorphism algebra over $\F_{3^{3}}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \). |
| The base change of $A$ to $\F_{3^{6}}$ is the simple isogeny class 1.729.cc and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$. |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{3}}$.
| Subfield | Primitive Model |
| $\F_{3}$ | 1.3.ad |
| $\F_{3}$ | 1.3.a |
| $\F_{3}$ | 1.3.d |
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 1.27.aj | $3$ | (not in LMFDB) |
| 1.27.j | $3$ | (not in LMFDB) |
| 1.27.j | $6$ | (not in LMFDB) |