Invariants
Base field: | $\F_{2^{3}}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 5 x + 8 x^{2} )( 1 - 4 x + 8 x^{2} )( 1 - 3 x + 8 x^{2} )$ |
$1 - 12 x + 71 x^{2} - 252 x^{3} + 568 x^{4} - 768 x^{5} + 512 x^{6}$ | |
Frobenius angles: | $\pm0.154919815756$, $\pm0.250000000000$, $\pm0.322067999368$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $120$ | $262080$ | $154487880$ | $73115078400$ | $35625864852600$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-3$ | $63$ | $585$ | $4351$ | $33177$ | $262143$ | $2095881$ | $16776959$ | $134230905$ | $1073782143$ |
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 $\times$ 1.8.ad 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.db $\times$ 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 $\times$ 1.64.h. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.