Invariants
Base field: | $\F_{2^{2}}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - x + 4 x^{2} )( 1 - 3 x + 4 x^{2} )^{2}$ |
$1 - 7 x + 27 x^{2} - 65 x^{3} + 108 x^{4} - 112 x^{5} + 64 x^{6}$ | |
Frobenius angles: | $\pm0.230053456163$, $\pm0.230053456163$, $\pm0.419569376745$ |
Angle rank: | $2$ (numerical) |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $3$ |
Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $16$ | $6144$ | $416176$ | $19906560$ | $1128577936$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-2$ | $22$ | $94$ | $302$ | $1078$ | $4198$ | $16462$ | $64862$ | $259366$ | $1044502$ |
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 $\F_{2^{2}}$.
Endomorphism algebra over $\F_{2^{2}}$The isogeny class factors as 1.4.ad 2 $\times$ 1.4.ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
|
Base change
This is a primitive isogeny class.