Invariants
| Base field: | $\F_{2^{2}}$ |
| Dimension: | $4$ |
| L-polynomial: | $( 1 - 2 x )^{4}( 1 - 3 x + 4 x^{2} )( 1 + x + 4 x^{2} )$ |
| $1 - 10 x + 45 x^{2} - 128 x^{3} + 280 x^{4} - 512 x^{5} + 720 x^{6} - 640 x^{7} + 256 x^{8}$ | |
| Frobenius angles: | $0$, $0$, $0$, $0$, $\pm0.230053456163$, $\pm0.580430623255$ |
| Angle rank: | $2$ (numerical) |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $12$ | $31104$ | $9594396$ | $3499200000$ | $1085185198092$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-5$ | $7$ | $31$ | $207$ | $1015$ | $3895$ | $15535$ | $64287$ | $259879$ | $1041607$ |
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.ae 2 $\times$ 1.4.ad $\times$ 1.4.b 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.