Invariants
Base field: | $\F_{2}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 2 x + 2 x^{2} )^{3}$ |
$1 - 6 x + 18 x^{2} - 32 x^{3} + 36 x^{4} - 24 x^{5} + 8 x^{6}$ | |
Frobenius angles: | $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$ |
Angle rank: | $0$ (numerical) |
Jacobians: | $0$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1$ | $125$ | $2197$ | $15625$ | $68921$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-3$ | $5$ | $21$ | $41$ | $57$ | $65$ | $81$ | $161$ | $417$ | $1025$ |
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 $\F_{2^{4}}$.
Endomorphism algebra over $\F_{2}$The isogeny class factors as 1.2.ac 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$ |
The base change of $A$ to $\F_{2^{4}}$ is 1.16.i 3 and its endomorphism algebra is $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$. |
- Endomorphism algebra over $\F_{2^{2}}$
The base change of $A$ to $\F_{2^{2}}$ is 1.4.a 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$
Base change
This is a primitive isogeny class.