Invariants
| Base field: | $\F_{2^{2}}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - 2 x )^{2}( 1 - x - 3 x^{2} - 4 x^{3} + 16 x^{4} )$ |
| $1 - 5 x + 5 x^{2} + 4 x^{3} + 20 x^{4} - 80 x^{5} + 64 x^{6}$ | |
| Frobenius angles: | $0$, $0$, $\pm0.0862360434115$, $\pm0.752902710078$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $9$ | $1539$ | $142884$ | $15736275$ | $950187789$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $0$ | $2$ | $27$ | $242$ | $900$ | $3983$ | $16380$ | $64802$ | $262683$ | $1048202$ |
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 $\times$ 2.4.ab_ad 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 $\times$ 1.64.al 2 . The endomorphism algebra for each factor is:
|
Base change
This is a primitive isogeny class.