Invariants
| Base field: | $\F_{2^{2}}$ |
| Dimension: | $4$ |
| L-polynomial: | $( 1 - 3 x + 4 x^{2} )^{2}( 1 - 4 x + 9 x^{2} - 16 x^{3} + 16 x^{4} )$ |
| $1 - 10 x + 50 x^{2} - 162 x^{3} + 377 x^{4} - 648 x^{5} + 800 x^{6} - 640 x^{7} + 256 x^{8}$ | |
| Frobenius angles: | $\pm0.117169895439$, $\pm0.230053456163$, $\pm0.230053456163$, $\pm0.478661301576$ |
| Angle rank: | $3$ (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: | $4$ |
| Slopes: | $[0, 0, 0, 0, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $24$ | $70656$ | $21586392$ | $4761649152$ | $1224771644184$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-5$ | $17$ | $79$ | $285$ | $1135$ | $4385$ | $16543$ | $64701$ | $260575$ | $1049537$ |
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 $\times$ 2.4.ae_j 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.