Invariants
| Base field: | $\F_{3^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 3 x )^{4}$ |
| $1 - 12 x + 54 x^{2} - 108 x^{3} + 81 x^{4}$ | |
| Frobenius angles: | $0$, $0$, $0$, $0$ |
| Angle rank: | $0$ (numerical) |
| Jacobians: | $0$ |
This isogeny class is not simple, not primitive, not ordinary, and supersingular. It is principally polarizable.
Newton polygon
This isogeny class is supersingular.
| $p$-rank: | $0$ |
| Slopes: | $[1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $16$ | $4096$ | $456976$ | $40960000$ | $3429742096$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-2$ | $46$ | $622$ | $6238$ | $58078$ | $528526$ | $4774222$ | $43020478$ | $387341758$ | $3486548206$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{2}}$.
Endomorphism algebra over $\F_{3^{2}}$| The isogeny class factors as 1.9.ag 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ 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 $\F_{3^{2}}$.
| Subfield | Primitive Model |
| $\F_{3}$ | 2.3.a_ag |