Invariants
Base field: | $\F_{2^{2}}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 + 4 x^{2} )( 1 - 3 x + 4 x^{2} )^{2}$ |
$1 - 6 x + 21 x^{2} - 48 x^{3} + 84 x^{4} - 96 x^{5} + 64 x^{6}$ | |
Frobenius angles: | $\pm0.230053456163$, $\pm0.230053456163$, $\pm0.5$ |
Angle rank: | $1$ (numerical) |
This isogeny class is not simple, not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $20$ | $6400$ | $355940$ | $18662400$ | $1199992100$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-1$ | $23$ | $83$ | $287$ | $1139$ | $4319$ | $16211$ | $64127$ | $260147$ | $1048223$ |
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 $\F_{2^{2}}$The isogeny class factors as 1.4.ad 2 $\times$ 1.4.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.ab 2 $\times$ 1.16.i. The endomorphism algebra for each factor is:
|
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{2}}$.
Subfield | Primitive Model |
$\F_{2}$ | 3.2.ac_ab_g |
$\F_{2}$ | 3.2.c_ab_ag |