Invariants
| Base field: | $\F_{2^{2}}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - 2 x )^{2}( 1 - 3 x + 4 x^{2} )( 1 - x + 4 x^{2} )$ |
| $1 - 8 x + 31 x^{2} - 76 x^{3} + 124 x^{4} - 128 x^{5} + 64 x^{6}$ | |
| Frobenius angles: | $0$, $0$, $\pm0.230053456163$, $\pm0.419569376745$ |
| 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: | $[0, 0, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $8$ | $3456$ | $275576$ | $15552000$ | $1002369128$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-3$ | $15$ | $69$ | $239$ | $957$ | $4023$ | $16293$ | $64799$ | $259341$ | $1043655$ |
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 $\times$ 1.4.ad $\times$ 1.4.ab 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.