Invariants
Base field: | $\F_{2^{4}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 7 x + 16 x^{2} )( 1 + 16 x^{2} )$ |
$1 - 7 x + 32 x^{2} - 112 x^{3} + 256 x^{4}$ | |
Frobenius angles: | $\pm0.160861246510$, $\pm0.5$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $4$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $170$ | $69360$ | $16756730$ | $4276044000$ | $1101267994250$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $10$ | $272$ | $4090$ | $65248$ | $1050250$ | $16793552$ | $268465690$ | $4294917568$ | $68719562410$ | $1099513023152$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+xy=a^3x^5+a^2x^3+a^3x^2+x$
- $y^2+xy=(a^3+a^2)x^5+(a+1)x^3+(a^3+1)x^2+x$
- $y^2+xy=(a^3+a)x^5+ax^3+(a^3+a^2+a)x^2+x$
- $y^2+xy=(a^3+a^2+a+1)x^5+(a^2+1)x^3+(a^3+a^2+a+1)x^2+x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{8}}$.
Endomorphism algebra over $\F_{2^{4}}$The isogeny class factors as 1.16.ah $\times$ 1.16.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 $\F_{2^{8}}$ is 1.256.ar $\times$ 1.256.bg. The endomorphism algebra for each factor is:
|
Base change
This is a primitive isogeny class.