Invariants
Base field: | $\F_{2}$ |
Dimension: | $6$ |
L-polynomial: | $( 1 + 2 x^{2} )( 1 - 2 x + 2 x^{2} )^{5}$ |
$1 - 10 x + 52 x^{2} - 180 x^{3} + 460 x^{4} - 912 x^{5} + 1440 x^{6} - 1824 x^{7} + 1840 x^{8} - 1440 x^{9} + 832 x^{10} - 320 x^{11} + 64 x^{12}$ | |
Frobenius angles: | $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.5$ |
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: | $[1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $3$ | $28125$ | $3341637$ | $87890625$ | $3823254633$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-7$ | $9$ | $29$ | $49$ | $73$ | $81$ | $49$ | $65$ | $353$ | $1089$ |
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 $\F_{2^{8}}$.
Endomorphism algebra over $\F_{2}$The isogeny class factors as 1.2.ac 5 $\times$ 1.2.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
|
The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg 6 and its endomorphism algebra is $\mathrm{M}_{6}(B)$, 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 5 $\times$ 1.4.e. The endomorphism algebra for each factor is: - 1.4.a 5 : $\mathrm{M}_{5}($\(\Q(\sqrt{-1}) \)$)$
- 1.4.e : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- Endomorphism algebra over $\F_{2^{4}}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.ai $\times$ 1.16.i 5 . The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.