Invariants
| Base field: | $\F_{2^{2}}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - 2 x )^{4}( 1 + 3 x + 4 x^{2} )$ |
| $1 - 5 x + 4 x^{2} + 8 x^{3} + 16 x^{4} - 80 x^{5} + 64 x^{6}$ | |
| Frobenius angles: | $0$, $0$, $0$, $0$, $\pm0.769946543837$ |
| 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: | $1$ |
| Slopes: | $[0, 1/2, 1/2, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $8$ | $1296$ | $134456$ | $14580000$ | $893968328$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $0$ | $0$ | $24$ | $224$ | $840$ | $3888$ | $15960$ | $64064$ | $261096$ | $1043280$ |
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.d 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 isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{2}}$.
| Subfield | Primitive Model |
| $\F_{2}$ | 3.2.ab_ac_e |
| $\F_{2}$ | 3.2.b_ac_ae |