Invariants
Base field: | $\F_{2^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 4 x + 8 x^{2} )( 1 + x + 8 x^{2} )$ |
$1 - 3 x + 12 x^{2} - 24 x^{3} + 64 x^{4}$ | |
Frobenius angles: | $\pm0.250000000000$, $\pm0.556567041129$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $3$ |
This isogeny class is not simple, 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $50$ | $5200$ | $267050$ | $16900000$ | $1091476250$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $80$ | $522$ | $4128$ | $33306$ | $262640$ | $2092362$ | $16767808$ | $134224506$ | $1073728400$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+xy=(a+1)x^5+(a^2+a)x^3+x^2+x$
- $y^2+xy=(a^2+a+1)x^5+a^2x^3+x^2+x$
- $y^2+xy=(a^2+1)x^5+ax^3+x^2+x$
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 $\times$ 1.8.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 $\F_{2^{12}}$ is 1.4096.adt $\times$ 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.a $\times$ 1.64.p. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.