Invariants
Base field: | $\F_{2^{2}}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 2 x + 4 x^{2} )^{3}$ |
$1 - 6 x + 24 x^{2} - 56 x^{3} + 96 x^{4} - 96 x^{5} + 64 x^{6}$ | |
Frobenius angles: | $\pm0.333333333333$, $\pm0.333333333333$, $\pm0.333333333333$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $27$ | $9261$ | $531441$ | $20346417$ | $979146657$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-1$ | $29$ | $113$ | $305$ | $929$ | $3713$ | $16001$ | $66305$ | $265217$ | $1051649$ |
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 $\F_{2^{6}}$.
Endomorphism algebra over $\F_{2^{2}}$The isogeny class factors as 1.4.ac 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$ |
The base change of $A$ to $\F_{2^{6}}$ is 1.64.q 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.