Invariants
Base field: | $\F_{2}$ |
Dimension: | $6$ |
L-polynomial: | $( 1 - x + 2 x^{2} )( 1 - 2 x + 2 x^{2} )^{3}( 1 - 2 x + 3 x^{2} - 4 x^{3} + 4 x^{4} )$ |
$1 - 9 x + 43 x^{2} - 139 x^{3} + 338 x^{4} - 650 x^{5} + 1016 x^{6} - 1300 x^{7} + 1352 x^{8} - 1112 x^{9} + 688 x^{10} - 288 x^{11} + 64 x^{12}$ | |
Frobenius angles: | $\pm0.174442860055$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.384973271919$, $\pm0.546783656212$ |
Angle rank: | $3$ (numerical) |
Jacobians: | $0$ |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $3$ |
Slopes: | $[0, 0, 0, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $4$ | $28000$ | $1906996$ | $56000000$ | $2671653644$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-6$ | $10$ | $24$ | $38$ | $64$ | $82$ | $92$ | $190$ | $456$ | $930$ |
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 $\times$ 1.2.ab $\times$ 2.2.ac_d 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 $\F_{2^{4}}$ is 1.16.ab $\times$ 1.16.i 3 $\times$ 2.16.ac_b. The endomorphism algebra for each factor is:
|
- Endomorphism algebra over $\F_{2^{2}}$
The base change of $A$ to $\F_{2^{2}}$ is 1.4.a 3 $\times$ 1.4.d $\times$ 2.4.c_b. The endomorphism algebra for each factor is: - 1.4.a 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$
- 1.4.d : \(\Q(\sqrt{-7}) \).
- 2.4.c_b : 4.0.1088.2.
Base change
This is a primitive isogeny class.