Invariants
Base field: | $\F_{2^{2}}$ |
Dimension: | $4$ |
L-polynomial: | $( 1 - 2 x )^{4}( 1 - 3 x + 5 x^{2} - 12 x^{3} + 16 x^{4} )$ |
$1 - 11 x + 53 x^{2} - 156 x^{3} + 344 x^{4} - 624 x^{5} + 848 x^{6} - 704 x^{7} + 256 x^{8}$ | |
Frobenius angles: | $0$, $0$, $0$, $0$, $\pm0.103279877171$, $\pm0.563386789496$ |
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: | $[0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $7$ | $20979$ | $7529536$ | $2950171875$ | $1024394428267$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-6$ | $2$ | $15$ | $162$ | $954$ | $3935$ | $15786$ | $64962$ | $262095$ | $1045682$ |
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.ae 2 $\times$ 2.4.ad_f 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.aq 2 $\times$ 1.64.aj 2 . 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 |
$\F_{2}$ | 4.2.ab_af_c_m |
$\F_{2}$ | 4.2.b_af_ac_m |