Invariants
Base field: | $\F_{2^{2}}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 2 x )^{2}( 1 - 2 x + 4 x^{2} )( 1 - x + 4 x^{2} )$ |
$1 - 7 x + 26 x^{2} - 64 x^{3} + 104 x^{4} - 112 x^{5} + 64 x^{6}$ | |
Frobenius angles: | $0$, $0$, $\pm0.333333333333$, $\pm0.419569376745$ |
Angle rank: | $1$ (numerical) |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $12$ | $4536$ | $301644$ | $14742000$ | $919919172$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-2$ | $20$ | $76$ | $224$ | $868$ | $3848$ | $16252$ | $65504$ | $261364$ | $1045880$ |
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 $\F_{2^{2}}$The isogeny class factors as 1.4.ae $\times$ 1.4.ac $\times$ 1.4.ab 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.aey 2 $\times$ 1.4096.h. The endomorphism algebra for each factor is:
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- Endomorphism algebra over $\F_{2^{4}}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.ai $\times$ 1.16.e $\times$ 1.16.h. The endomorphism algebra for each factor is: - 1.16.ai : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 1.16.e : \(\Q(\sqrt{-3}) \).
- 1.16.h : \(\Q(\sqrt{-15}) \).
- Endomorphism algebra over $\F_{2^{6}}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.aq $\times$ 1.64.l $\times$ 1.64.q. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.