Invariants
Base field: | $\F_{2}$ |
Dimension: | $6$ |
L-polynomial: | $( 1 - 2 x + 2 x^{2} )^{2}( 1 - 2 x + 2 x^{2} - 4 x^{3} + 4 x^{4} )^{2}$ |
$1 - 8 x + 32 x^{2} - 88 x^{3} + 192 x^{4} - 352 x^{5} + 544 x^{6} - 704 x^{7} + 768 x^{8} - 704 x^{9} + 512 x^{10} - 256 x^{11} + 64 x^{12}$ | |
Frobenius angles: | $\pm0.0833333333333$, $\pm0.0833333333333$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.583333333333$, $\pm0.583333333333$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1$ | $4225$ | $105625$ | $17850625$ | $2933413921$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-5$ | $5$ | $1$ | $17$ | $65$ | $65$ | $65$ | $257$ | $577$ | $1025$ |
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 $\F_{2^{12}}$.
Endomorphism algebra over $\F_{2}$The isogeny class factors as 1.2.ac 2 $\times$ 2.2.ac_c 2 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 $\F_{2^{12}}$ is 1.4096.ey 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 2 $\times$ 2.4.a_ae 2 . The endomorphism algebra for each factor is: - 1.4.a 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- 2.4.a_ae 2 : $\mathrm{M}_{2}($\(\Q(\zeta_{12})\)$)$
- Endomorphism algebra over $\F_{2^{3}}$
The base change of $A$ to $\F_{2^{3}}$ is 1.8.ae 4 $\times$ 1.8.e 2 . The endomorphism algebra for each factor is: - 1.8.ae 4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-1}) \)$)$
- 1.8.e 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- Endomorphism algebra over $\F_{2^{4}}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.ae 4 $\times$ 1.16.i 2 . The endomorphism algebra for each factor is: - 1.16.ae 4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-3}) \)$)$
- 1.16.i 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- Endomorphism algebra over $\F_{2^{6}}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.a 6 and its endomorphism algebra is $\mathrm{M}_{6}($\(\Q(\sqrt{-1}) \)$)$
Base change
This is a primitive isogeny class.