Invariants
Base field: | $\F_{2^{2}}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 2 x + 4 x^{2} )( 1 - 3 x + 4 x^{2} )^{2}$ |
$1 - 8 x + 33 x^{2} - 82 x^{3} + 132 x^{4} - 128 x^{5} + 64 x^{6}$ | |
Frobenius angles: | $\pm0.230053456163$, $\pm0.230053456163$, $\pm0.333333333333$ |
Angle rank: | $1$ (numerical) |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $12$ | $5376$ | $443556$ | $22643712$ | $1162528932$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-3$ | $19$ | $99$ | $335$ | $1107$ | $4063$ | $16083$ | $64895$ | $261171$ | $1047199$ |
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 $\F_{2^{6}}$.
Endomorphism algebra over $\F_{2^{2}}$The isogeny class factors as 1.4.ad 2 $\times$ 1.4.ac 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 $\F_{2^{6}}$ is 1.64.j 2 $\times$ 1.64.q. The endomorphism algebra for each factor is:
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Base change
This is a primitive isogeny class.