Invariants
Base field: | $\F_{2^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 5 x + 8 x^{2} )( 1 - 4 x + 8 x^{2} )$ |
$1 - 9 x + 36 x^{2} - 72 x^{3} + 64 x^{4}$ | |
Frobenius angles: | $\pm0.154919815756$, $\pm0.250000000000$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $0$ |
This isogeny class is not simple, not primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $20$ | $3640$ | $276860$ | $17508400$ | $1091278100$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $0$ | $56$ | $540$ | $4272$ | $33300$ | $263144$ | $2097900$ | $16775008$ | $134208900$ | $1073731736$ |
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.af $\times$ 1.8.ae 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.bv $\times$ 1.4096.ey. 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.aj $\times$ 1.64.a. 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 $\F_{2^{3}}$.
Subfield | Primitive Model |
$\F_{2}$ | 2.2.d_g |