Invariants
Base field: | $\F_{2^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 5 x + 8 x^{2} )( 1 + 8 x^{2} )$ |
$1 - 5 x + 16 x^{2} - 40 x^{3} + 64 x^{4}$ | |
Frobenius angles: | $\pm0.154919815756$, $\pm0.5$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $3$ |
This isogeny class is not simple, not primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $36$ | $4536$ | $260604$ | $16447536$ | $1082818836$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $72$ | $508$ | $4016$ | $33044$ | $264168$ | $2099948$ | $16775008$ | $134225284$ | $1073797272$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+xy=(a+1)x^5+(a^2+a+1)x^3+x$
- $y^2+xy=(a^2+a+1)x^5+(a^2+1)x^3+x$
- $y^2+xy=(a^2+1)x^5+(a+1)x^3+x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{6}}$.
Endomorphism algebra over $\F_{2^{3}}$The isogeny class factors as 1.8.af $\times$ 1.8.a 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 $\F_{2^{6}}$ is 1.64.aj $\times$ 1.64.q. 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.b_e |