Invariants
Base field: | $\F_{2^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 2 x )^{2}( 1 + 4 x^{2} )$ |
$1 - 4 x + 8 x^{2} - 16 x^{3} + 16 x^{4}$ | |
Frobenius angles: | $0$, $0$, $\pm0.5$ |
Angle rank: | $0$ (numerical) |
Jacobians: | $1$ |
This isogeny class is not simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is supersingular.
$p$-rank: | $0$ |
Slopes: | $[1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $5$ | $225$ | $3185$ | $50625$ | $985025$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $1$ | $17$ | $49$ | $193$ | $961$ | $4097$ | $16129$ | $64513$ | $261121$ | $1048577$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2+y=x^5+x^4+a$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{8}}$.
Endomorphism algebra over $\F_{2^{2}}$The isogeny class factors as 1.4.ae $\times$ 1.4.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.abg 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$. |
- Endomorphism algebra over $\F_{2^{4}}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.ai $\times$ 1.16.i. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.