Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 + x + 2 x^{2} )( 1 - x + 2 x^{2} )^{2}$ |
| $1 - x + 5 x^{2} - 3 x^{3} + 10 x^{4} - 4 x^{5} + 8 x^{6}$ | |
| Frobenius angles: | $\pm0.384973271919$, $\pm0.384973271919$, $\pm0.615026728081$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $0$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $16$ | $512$ | $784$ | $4096$ | $21296$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $2$ | $14$ | $14$ | $14$ | $22$ | $38$ | $142$ | $350$ | $518$ | $854$ |
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 $\F_{2}$The isogeny class factors as 1.2.ab 2 $\times$ 1.2.b and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
|
| The base change of $A$ to $\F_{2^{2}}$ is 1.4.d 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$ |
Base change
This is a primitive isogeny class.