Invariants
| Base field: | $\F_{2^{3}}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - 4 x + 8 x^{2} )( 1 - 6 x + 23 x^{2} - 48 x^{3} + 64 x^{4} )$ |
| $1 - 10 x + 55 x^{2} - 188 x^{3} + 440 x^{4} - 640 x^{5} + 512 x^{6}$ | |
| Frobenius angles: | $\pm0.215051068352$, $\pm0.250000000000$, $\pm0.409556076405$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $170$ | $313820$ | $158894750$ | $71965202400$ | $35451834000850$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-1$ | $75$ | $599$ | $4287$ | $33019$ | $262635$ | $2097703$ | $16769087$ | $134165867$ | $1073620875$ |
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 $\F_{2^{12}}$.
Endomorphism algebra over $\F_{2^{3}}$| The isogeny class factors as 1.8.ae $\times$ 2.8.ag_x 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 $\F_{2^{12}}$ is 1.4096.ey $\times$ 2.4096.ck_cxd. The endomorphism algebra for each factor is:
|
- Endomorphism algebra over $\F_{2^{6}}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.a $\times$ 2.64.k_dd. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.