Invariants
Base field: | $\F_{2}$ |
Dimension: | $6$ |
L-polynomial: | $( 1 - 2 x + 2 x^{2} )^{3}( 1 - 2 x + x^{2} + 2 x^{4} - 8 x^{5} + 8 x^{6} )$ |
$1 - 8 x + 31 x^{2} - 74 x^{3} + 120 x^{4} - 148 x^{5} + 184 x^{6} - 296 x^{7} + 480 x^{8} - 592 x^{9} + 496 x^{10} - 256 x^{11} + 64 x^{12}$ | |
Frobenius angles: | $\pm0.0693533547550$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.339907131295$, $\pm0.770553776540$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $0$ |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2$ | $5500$ | $848042$ | $93500000$ | $1153875382$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-5$ | $3$ | $19$ | $47$ | $35$ | $51$ | $79$ | $159$ | $523$ | $1043$ |
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 $\times$ 3.2.ac_b_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 $\F_{2^{4}}$ is 1.16.i 3 $\times$ 3.16.g_r_ce. The endomorphism algebra for each factor is:
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- Endomorphism algebra over $\F_{2^{2}}$
The base change of $A$ to $\F_{2^{2}}$ is 1.4.a 3 $\times$ 3.4.ac_f_am. The endomorphism algebra for each factor is: - 1.4.a 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$
- 3.4.ac_f_am : 6.0.2580992.1.
Base change
This is a primitive isogeny class.