Invariants
| Base field: | $\F_{2^{2}}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - 2 x )^{6}$ |
| $1 - 12 x + 60 x^{2} - 160 x^{3} + 240 x^{4} - 192 x^{5} + 64 x^{6}$ | |
| Frobenius angles: | $0$, $0$, $0$, $0$, $0$, $0$ |
| Angle rank: | $0$ (numerical) |
This isogeny class is not simple, primitive, not ordinary, and supersingular. It is principally polarizable.
Newton polygon
This isogeny class is supersingular.
| $p$-rank: | $0$ |
| Slopes: | $[1/2, 1/2, 1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1$ | $729$ | $117649$ | $11390625$ | $887503681$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-7$ | $-7$ | $17$ | $161$ | $833$ | $3713$ | $15617$ | $64001$ | $259073$ | $1042433$ |
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 $\F_{2^{2}}$.
Endomorphism algebra over $\F_{2^{2}}$| The isogeny class factors as 1.4.ae 3 and its endomorphism algebra is $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$. |
Base change
This is a primitive isogeny class.