Invariants
| Base field: | $\F_{2^{2}}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - 3 x + 4 x^{2} )( 1 - 4 x + 9 x^{2} - 16 x^{3} + 16 x^{4} )$ |
| $1 - 7 x + 25 x^{2} - 59 x^{3} + 100 x^{4} - 112 x^{5} + 64 x^{6}$ | |
| Frobenius angles: | $\pm0.117169895439$, $\pm0.230053456163$, $\pm0.478661301576$ |
| Angle rank: | $3$ (numerical) |
| Isomorphism classes: | 8 |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $3$ |
| Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $12$ | $4416$ | $291708$ | $16533504$ | $1131951612$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-2$ | $18$ | $70$ | $254$ | $1078$ | $4338$ | $16630$ | $65150$ | $261574$ | $1050738$ |
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 $\times$ 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.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 3.4.ab_b_af | $2$ | 3.16.b_ab_db |
| 3.4.b_b_f | $2$ | 3.16.b_ab_db |
| 3.4.h_z_ch | $2$ | 3.16.b_ab_db |